A Simple Version of the Lucas Model

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1 Aricle non publié May 11, 2007 A Simple Version of he Lucas Model Mazamba Tédie Absrac This discree-ime version of he Lucas model do no include he physical capial. We inregrae in he uiliy funcion he leisure ime. We examine he social planer and he compeiive equilibrium. The main conclusions are ha he consumer always chooses o rain, he human capial growh rae increases wih he exernaliy and he qualiy of raining, and ha he equilibrium defined by Lucas 1988 is a compeiive equilibrium under some condiions. 1. Inroducion This model is a discree-ime version of he model of Lucas wihou physical capial. The consumer devoes he fracion θ of his non-leisure ime o curren producion and he remaining 1 θ o human capial accumulaion. We consider ha he uiliy of consumer increases wih his leisure ime. This assumpion implies ha he uiliy increases wih he human capial accumulaion ha is wih he raining. Following Lucas 1988, he human capial has : 1- an exernal effec hrough he exernaliy. 2- an inernal effec which increases he produciviy hrough he medium of raining. This paper is organized ino seven secions. Secion 2 inroduces assumpions and examines he social planer problem. Afer defining he equilibrium according o Lucas and Romer and compeiive equilibrium, secion 3 shows ha an equilibrium is a compeiive equilibrium. The following secions conclude and give some proofs. 2. Social Planer The uiliy funcion is concave 0 < µ < 1 and 0 < ζ < 1 : Subjec o, max β c µ 1 θ ζ The auhor hanks Gourdel, P. Universiy of Paris 1, CERMSEM & Le Van, C. Universiy of Paris 1, CNRS-CERMSEM for helpful commens during he course of his presenaion. He akes responsabiliy for any remaining errors Keywords : Human capial, Economic Growh, Compeiive Equilibrium, Equilibrium 1

2 We make he following assumpions : 0, 0 c h γ θ h α h +1 = h 1 + λφ1 θ 0 < α < 1, γ 0, 0 θ 1, > 0 given H1 : φ is concave, increasing and wice coninuously differeniable. φ0 = 0, φ1 = 1, λ > 0 and φ 0 > 1. H2 : 0 < β1 + λ α+γµ < 1. The parameer λ balanced he echnology of raining. Le us define he funcion ψ : [1, 1 + λ] [0, 1] by ψx = 1 φ 1 1 x 1. Where φ 1 denoes he inverse funcion of φ. ψ is clearly decreasing. I is easy o verify ha : ψ1 = 1 and ψ1 + λ = 0. λ This funcion gives he working ime when he human capial grows by facor x. ψ is coninuously differeniable, decreasing, wih ψ 1 = 1, λφ 0 ψ 1 + λ = 1 and λφ 1 concave. The problem becomes : Subjec o : max β h α+γµ ψ h αµ +1 1 ψ h ζ +1 h h 0, h h +1 h 1 + λ and > 0 given. Proposiion 1 Under H1-H2, here exiss a soluion. Proof. See he appendix 1. Proposiion 2 Each opimal pah of human capial h =, h 1,..., h,... verifies < h 1 < < h <. Proof. See he appendix 2. Proposiion 3 Under assumpions H1,H2 and H3 : α + γµ 1 < 0 : a The opimal pah of human capial has a consan growh rae, sricly posiive and which increases wih parameer γ. b The opimal pah of human capial is an increasing funcion of λ. Proof. We prove a in several sages. 1. Le V be he Value Funcion of our problem of opimal growh : V = max β h α+γµ ψ h αµ +1 1 ψ h ζ +1 h h Under he consrains : 0, h h +1 h 1 + λ, and > 0 given. This value funcion saisfied see Le Van & Morhaim 2002 : 2

3 V = Ah α+γµ 0 Le us consider, he opimal value h 1 of he human capial a dae 1 is he soluion of he following equaion : { αµ ζ } α+γµ h α+γµ 0 max y [h0,1+λ ] ψ y 1 ψ y + βa y We can see ha h 1 = ν where ν is he soluion of he equaion : max { ψz αµ 1 ψz ζ + βaγz α+γµ}. Since he problem is saionary, if {h } is he opimal pah, hen : h = ν,. 2. We know ha he human capial pah verifies h +1 > h, 0. The Euler equaion is given by : h α+γµ 1 ψ h +1 h Ψ h +1 h [ αµ = βα + γµh α+γµ 1 +1 Ψ h +2 h +1 +βh α+γµ 1 h h +1 ψ h +2 h +1 Ψ h +2 h +1 ] 1 ψ h +1 h ζψ h +1 h ψ h +2 h +1 1 ψ h +2 h +1 [ αµ 1 ψ h +2 h +1 ζψ h +2 h +1 αµ 1 ζ 1. Wih Ψ h +1 h = ψ h +1 h 1 ψ h +1 h This equaion gives he human capial growh rae ha is consan ν : 1 = ζ ψν αµ 1 ψν 1 βν α+γµ β α + α γνα+γµ 1 ψν + ψ ν βνα+γµ Le F ν = ζ ψν αµ 1 ψν 1 βν α+γµ wih Gν = β α + α γνα+γµ ψν + ψ ν βνα+γµ. Funcions F and G are decreasing since : F x = 1 βx α+γµ ζ ψ x βα+γ ψx αµ 1 ψx 2 α 1 ψx xα+γµ 1 < 0, G x = β [ α + α γx α+γµ 2 ψx α + γµ 1 ψ x x + x 1 µ] < 0. Moreover, F 1+λ = ψ x ψx α 0, lim x 1 F x = +, G1 = βα+γ λφ 0 and G1 + λ = β1 + λ α+γµ < 1 according o H2. Hence, here exiss a unique soluion ν ]1, 1 + λ[. α 3. We know ha he value funcion verifies he Bellman equaion : V h = h α+γµ max ν [1,1+λ] { ψν αµ 1 ψν ζ + βaγν α+γµ} The derivae of funcion ψν αµ 1 ψν ζ + βaγν α+γµ is cancelled : αµ ψ ν ψν ζ + ζ ν ψν αµ 1 ψν ζ 1 = βaγα + γµν α+γµ 1 When γ increases, he graph of he funcion βaγα + γµν α+γµ 1 moves o he op while he lef-hand side remains consan. Consequenly, he growh rae increases wih he parameer of he exernaliy. This ends he proof of he claim a. 4. Le us rewrie he Euler equaion : 1 = F λ x + G λ x. Noe ha λ < λ ψ λ < ψ λ and ψ λ < ψ λ. Hence, F and G are increasing wih λ. Moreover, F and G are decreasing wih ν, hen : [ dν F dλ = λ + G G / λ ν + F ] > 0 ν 3 ]

4 3. Equilibrium and Compeiive Equilibrium We inroduce he conceps of equilibrium according o Lucas and Romer and compeiive equilibrium. Take a human capial pah h = h 1,..., h,... o be given. Given h, consider he problem : Under he consrains, max c β uc, θ, 0 c G hfθ h h +1 = h 1 + λφ1 θ 0 θ 1, > 0 given The soluion h =, h 1,..., h,... of his model depends on h. In ohers words, h = Φ h. A equilibrium is a human capial pah h =,...,,... such ha h = Φh. In order o define a compeiive equilibrium, we need before o define he space of he prices which suppors his equilibrium. Observe ha all feasible pahs of consumpion c verify for all : 0 c h α+γ wih h 1 + λ. In ohers words, c belongs o : { } l c = c : sup < λ α+γ,..,+ Le l + be he se of non negaive sequences of l. The price sequence p is such as all consumpion pahs c verify + p c < +. Likewise, he wage pah w is such as w h < +. In order o saisfy hese wo condiions, we mus ake he prices space and he wages space as follows : { l 1 p = p : p 1 + λ α+γ < + } { ; l 1 w = w : w 1 + λ < + } Le us denoe l 1 +, he se of non-negaive sequences of l 1. We define a compeiive equilibrium for he model of Lucas. A collecion of sequences h, c, θ, p, w is a compeiive equilibrium if : 1. c, θ is a soluion of he consumer program : Under he consrains, max c,θ p c β uc c, θ 0, θ = ψ h +1 h 2. θ is a soluion of he firm program : 4 w θ h + Π, > 0 given

5 { + Π = max p γ θ α θ 3. Equilibrium on he goods and services marke : w θ } 0, c = γ θ α Proposiion 4 h is a equilibrium from > 0 if and only if i verifies he hree following condiions : 1. Inerioriy : 0, < +1 < 1 + λ, h 0 = > 0 2. Euler equaion 0, αµ h α+γµ 1 ψ h +1 ζ h α+γµ 1 ψ h +1 +βαµ h α+γµ 1 +1 βαµ h α+γµ βζ h α+γµ Transversaliy condiion, Proof. See he appendix 3. ψ h +1 ψ h lim + β h α+γµ 1 ψ h +1 αµ 1 αµ 1 ψ h +1 ζ ζ 1 ψ h +1 1 ψ h +1 αµ ζ 1 ψ h αµ 1 ζ ψ h +2 ψ h +2 1 ψ h αµ ζ 1 ψ h +2 ψ h +2 1 ψ h h = 0 +1 [ αµ ψ h +1 1 ψ h +1 αµ 1 ζ 1 1 ψ h +1 ] ζψ h +1 h +1 = 0 Proposiion 5 Under he assumpions of proposiion 3 and H4 : λ 1 1, here exiss β an equilibrium h which increases a consan rae ν. The equilibrium growh rae h is weaker han ha of he cenralized rae. We can associae wih his equilibrium he saionary sequence θ = ψν, a consumpion sequence c, a price sysem p, wage w such as he collecion of sequences h, c, θ, p, w is a compeiive equilibrium. Proof. 1. We know ha if h is an equilibrium hen i verifies inerioriy, he Euler equaion and he ransversaliy condiion. In addiion, le us show ha exiss a human capial sequence ha increases a consan rae and saisfies he Euler equaion. Indeed, according o Euler equaion, his rae ν mus saisfy : 1 = ζ ψν αµ 1 ψν 1 βν α+γµ βν α+γµ 1 ψν ψ ν + βνα+γµ V ν 5

6 Le F ν = ζ ψν αµ 1 ψν 1 βν α+γµ and Hν = βν α+γµ 1 ψν + ψ ν βνα+γµ. We know ha F is decreasing, lim x 1 F x = + and ha F 1 + λ = 0. We show ha H is also decreasing : ] H ν = βν [α α+γµ 2 + γµ 1 ψν ν ν ψνψ ν < 0. One has V x = ψ ν ψ ν 2 F x + Hx, V x = F x + H x, lim x 1 V x = lim x 1 F x + lim x 1 Hx = + and V 1 + λ = F 1 + λ + H1 + λ = β1 + λ α+γµ < 1 according o H2. Consequenly, here exiss a unique soluion ν which belongs o ]1, 1 + λ[. I s easy o show ha his rae is weaker han he rae of social planer program which is he soluion of he equaion : 1 = F ν + Gν, since Gν = Hν βγ α να+γµ 1 ψν. Le ψ ν h be he rajecory defined by : h 0 =, +1 = ν,. Obviously, i saisfies he inerioriy and Euler equaion. We mus show han i verifies he ransversaliy condiion o conclude ha h is an equilibirum. Now, β h α+γµ h αµ 1 ζ 1 +1 ψ h +1 ψ h +1 1 ψ h +1 h h A +1 = β h α+γµ 0 ν α+γµ ψ ννψν αµ 1 1 ψν ζ 1 Aν h α+γµ 0 ψν αµ 1 νψ ν1 ψν ζ 1 A [ β1 + λ α+γµ] Where A = αµ 1 ψ h +1 lim + β h α+γµ 1 ψ h +1 ζψ h +1. Assumpion H2 implies : αµ 1 ψ h +1 ζ 1 1 ψ h +1 A = 0 This is he ransversaliy condiion. 2. We show ha his rajecory is a compeiive equilibrium. Le us define he price pah and he wage pah, p, w by : p = β uc,θ c w = β h α+γµ 1 = µβ h α+γµ 1 ψν αµ 1 1 ψν ζ ψν αµ 1 1 ψν ζ 1 [αµ1 ψν ζψν] Where = ν. a I is easy o see ha he sequence θ defined by θ = ψν, for all, maximizes he profi of he enerprise according o p and w. b In order o prove ha he consumpion pah and he working ime pah c, θ maximize he consumer uiliy, consider : T T T = β uc, θ β uc, θ Since + β u c c = + w θ + Π and + β u c c < + w θ h + Π wih θ = ψh +1 /h, one has : T T β [h h +ζh α+γµ h +1 ζh α+γµ 1 ψ h +1 αµh α+γµ 1 αµ αµh α+γµ 1 ψ h +1 αµ 1 ψ h +1 ψ h +1 1 ψ h ψ h +1 Φ h +1 ζ 1 ψ h +1 αµ 1 1 ψ h +1 ] ζ 1 ψ h +1 ζ ψ h +1

7 αµ 1 Where Φ h +1 = ψ h +1 Using he Euler equaion, we obain : T β T h α+γµ 1 T [ αµ 1 ψ h T +1 1 ψ h +1 ψ h T +1 ζψ h T +1 αµ 1 ] ζ 1 ψ h T +1 ψ h +1 1 ψ h T +1 h +1 ζ 1 ψ h +1. = β T wt ψ h T +1 h +1 T By definiion of w T. According o he ransversaliy condiion, we conclude ha lim T + T 0. c The goods marke is balanced since for all : c = γ θ α. d To complee his proof, le us show ha p belongs o l 1 p and w belongs o l 1 w. One has : p 1 + λ α+γ < µb h α+γµ 1 0 [ β1 + λ α+γµ ] < + According o H2 and wih B = ψν αµ 1 1 ψν ζ. Likewise, w 1 + λ = C h α+γµ 1 0 [ β1 + λν α+γµ 1 ] < + According o H4, 1 < ν < 1 + λ and where C = ψν αµ 1 1 ψν ζ 1 [αµ1 ψν ζψν]. This ends he proof. A collecion of pahs h, c, θ, p, w is a compeiive equilibrium. 4. Conclusion This dicree-ime version of he Lucas model solves he social planer program and shows ha an equilibrium for his model is a compeiive equilibrium. Moreover, he model concludes ha : 1- when he uiliy depends on consumpion and leisure ime, he consumer always prefers o increase his skill level. 2- he qualiy of raining increases he human capial growh rae. 3- he exernaliy is relaed posiively o he human capial growh rae hrough i conribuion o he produciviy of all facors of producion. 5. Appendix 1 I s easy o verify ha if c = c 0, c 1,..., c,... is a feasible pah of consumpion, hen :, 0 c h α+γ λ α+γ. This shows ha all feasible pahs of consumpion are compac for his opology. Assumpion H2 ensures ha funcion : Uc = β uc, θ is coninuous for he produc opology. Exisence of he soluion rise from hese resuls. 7

8 6. Appendix 2 I s enough o show ha for any iniial condiions, > 0, he saionary pah,,...,,... is no opimal. Le ɛ > 0 be a sufficienly small number such as 1 + λφɛ 1 + λ and a pah h =, h 1,..., h,... which verify h = 1 + λφɛ, 1. The consumpion pah associaed wih his human capial pah is c ɛ = c 0ɛ, c 1ɛ,..., c ɛ,... ha is : c 0ɛ = h α+γ 0 1 ɛ α and c ɛ = h α+γ λφɛ α+γ, 1. Moreover, le,,...,,... be a human capial pah and c be a consumpion pah which saisfy : c = h α+γ 0. Compare he uiliies generaed by hese sequences of consumpions, we have : ɛ = β c µ ɛ 1 ψ h ζ +1 β c µ 1 ψ1 ζ h Since ψ1 = 1, so ɛ > 0. All opimal pahs of human capial are increasing. 7. Appendix 3 We give he proof of he Proposiion 4 in several sages. 1. Le h be an equilibrium. One can show ha any equilibrium is increasing, ha is +1 >, 0 Proceed as in he previous appendix. Moreover, since he uiliy funcion verifies he Inada condiion, he opimal consumpions are sricly posiive on each dae. Hence, +1 < 1 + λ, for all. This ends he firs par of he claim. I is easy o show ha h verifies he Euler equaion see Le Van & Dana Le us show now ha he ransversaliy condiion is saisfied. Le V h be he value funcion of his program, one has : Under he consrains, V h = max β uc, θ 0 c G fθ h h +1 = h 1 + λφ1 θ 0 θ 1, > 0 given One can verify ha V h is concave and differeniable Benevise & Scheinkman 1979 and : αµ ζ V h = αµh α+γµ 1 0 ψ h 1 1 ψ h 1 αµh α+γµ 1 h αµ 1 ζ 1 0 ψ h 1 ψ h 1 1 ψ h 1 +ζh α+γµ 1 h αµ ζ ψ h 1 ψ h 1 1 ψ h 1 Moreover, since h is a equilibrium, i mus verify λ for all. Consequenly, c [ 1 + λ ] α+γ and 0 V h = β uc, θ h α+γµ 0 [β1 + λ α+γµ ] 8

9 Like V h 0 = 0, we have for all : h α+γµ 1 β1 + λ α+γµ V h V h 0 V h h Since, V h h = αµ h α+γµ 1 αµ h α+γµ 1 +ζ h α+γµ 1 +1 ψ h +1 ψ h α+γµ 1 +1 ψ h α+γµ 1 αµ ψ h +1 ψ h +1 ζ 1 ψ h +1 αµ 1 1 ψ h +1 αµ 1 ψ h +1 ζ 1 ζ and 1 + λ. Muliply he wo previous equaions by β, we obain he ransversaliy condiion : The Euler condiion implies : lim + β h α+γµ 1 ψ h +1 [ αµ 1 ψ h +1 +ζ h +1 ψ h +1 ψ h +1 lim + β h α+γµ 1 ψ h +1 [ αµ αµ 1 ψ h +1 ψ h +1 1 ψ h +1 1 ψ h +1 h +1 1 ψ h +1 αµ 1 ζ 1 ψ h +1 h ] = 0 ζ 1 1 ψ h +1 ] +1 = 0 ζψ h We prove he converse now. Le c,h and c,h be wo sequences ses wih he α. same iniial condiion. The las verifies, 0 c h γ ψ h +1 h Show ha T β uc, θ T β uc, θ 0. Observe ha u, x, y xψ y and ψ are x concave funcions, hence : T T β [h h u 1c, θ c, h h +1 + u 2c, θ θ, h h +1 ] ++1 h +1 u 1c, θ c h +1, +1 + u 2c, θ θ h h +1 Where u 1c, θ = u c c, θ and u 2c, θ = u θ c, θ. By he Euler equaion, T β T h α+γµ 1 T [ αµ 1 ψ h T +1 ψ h T +1 αµ 1 ζψ h T +1 1 ψ h T +1 h ] T ζ 1 ψ h T +1 h +1 T The ransversaliy condiion is wrien lim T + T = 0 since ψ < 0. 9

10 References [1] Barro, R., & Lee,J "Inernaional comparisons of educaional aainmen" Journal of Moneary Economics 32, [2] Benhabib, J., & Spiegel, M "The role of human capial in economic developmen : Evidence from aggregae cross-counry daa" Journal of Moneary Economics 34, [3] Le Van, C. & Morhaim, L "Opimal Growh Models wih Bounded or Unbounded Reurns : a Unifying Approach" Journal of Economic Theory 105, [4] Le Van, C., Morhaim, L. & Dimaria, C "The discree ime version of he Romer Model" Economic Theory 20, [5] Lucas, R.Jr 1988 "On he mechanics of economic developmen" Journal of Moneary Economics 22, [6] Mankiw, G., Romer, D. & Weil, D "A Conribuion o he Empirics of Economic Growh" The Quaerly Journal of Economics 107 2, [7] Romer, P "Endogenuous Technological Change" Journal of Poliical Economics 5, S71-S102. [8] Romer, P "Increasing reurns and long-run growh" Journal of Poliical Economy 94, mazambaedie@gmail.com hp://mazambaedie.free.fr 10