Closed form solutions of Lucas Uzawa model with externalities via partial Hamiltonian approach. Some Clarifications
aa r X i v : . [ ec on . T H ] J u l Closed-form solutions of Lucas–Uzawa modelwith externalities via partial Hamiltonianapproach. Some Clarifications
Constantin Chilarescu
Laboratoire CLERSE Universit´e de Lille, FranceE-mail: [email protected]
Abstract
The main aim of this paper is to give some clarifications to therecent paper published in Computational and Applied Mathematicsby Naz and Chaudhry.
Keywords : Lucas–Uzawa model with externalities; Uniqueness of solutions;Partial Hamiltonian approach.
Mathematics Subject Classification : 37 N
40; 49 J
15; 91 B JEL Classifications : C61, C62, O41.
The two sectors model considered in the paper of Lucas (1988), in terms ofper capita quantities, is given by the following definition.
Definition 1.
The set of paths { k, h, c, u } is called an optimal solution if itsolves the following optimization problem: V = max u,c ∞ Z [ c ( t )] − σ − − σ e − ρt dt, (1)1 ubject to ˙ k ( t ) = γ [ k ( t )] β [ h ( t )] − β + θ [ u ( t )] − β − πk ( t ) − c ( t ) , ˙ h ( t ) = δ [1 − u ( t )] h ( t ) ,k = k (0) , h = h (0) , (2) where k is physical capital, h is human capital, c is the real per-capita con-sumption and u is the fraction of labor allocated to the production of physicalcapital, with k > and h > being given. β is the elasticity of outputwith respect to physical capital, ρ is a positive discount factor, the efficiencyparameters γ > and δ > represent the constant technological levels inthe good sector and, respectively in the education sector θ is a positive exter-nality parameter and σ − represents the constant elasticity of intertemporalsubstitution, and throughout this paper we suppose that σ = 1 and σ = β . The equations (2) give the resources constraints and initial values for thestate variables k and h . Of course, the two state variables and the twocontrol variables c and u are all functions of times, but when no confusionsare possible, we simply write k, h , c and u . To solve the problem (1) subjectto (2), we define the Hamiltonian function: H = c − σ − − σ + (cid:20) γk β (cid:16) h − β + θ − β u (cid:17) − β − πk − c (cid:21) λ + δ (1 − u ) hµ. The boundary conditions include initial values ( k , h ), and the transversalityconditions: lim t →∞ e − ρt λ ( t ) k ( t ) = 0 and lim t →∞ e − ρt µ ( t ) h ( t ) = 0 . Differentiating the Hamiltonian with respect to c , u , k and h , we obtain theollowing dynamical system that drives the economy over time. ˙ k = " γ (cid:18) h − β + θ − β uk (cid:19) − β − π k − c, ˙ h = δ [1 − u ] h, ˙ c = " − ρ + πσ + γβσ (cid:18) h − β + θ − β uk (cid:19) − β c, ˙ u = h ( δ + π )(1 − β )+ θδβ − ck + δ (1 − β + θ )1 − β u i u, ˙ λ = " ρ + π − γβ (cid:18) h − β + θ − β uk (cid:19) − β λ ˙ µ = h ρ − δ − θδ − β u i µ. (3)The alternative of the above model, obtained via Hiraguchi transform, wasanalyzed, first of all by Boucekkine and Ruiz-Tamarit (2008), and later byChilarescu (2011) and both papers proved, doubtless that the model possessesa unique solution. More clarifications on the uniqueness of solutions to themodel of Lucas can be found in a recent paper of Chilarescu (2018 a ) .The model of Lucas with externalities was studied by Hiraguchi (2009)and he proved in his paper that this model possesses a unique set of solutions.The method employed by Hiraguchi was that of hypergeometric functions.In a recent paper of Chilarescu (2018 b ), the same model of Lucas with ex-ternalities was completely solved, this time in a simpler manner, using onlyclassical mathematical tools. He also provided a proof of the existence anduniqueness of solutions.In the paper we comment, Naz and Chaudhry (2018) claim that theyfound two different set of solutions for the model of Lucas with externalities.The first set of solutions coincides exactly with those determine by Hiraguchiand later by Chilarescu. In the second set, the solutions for the variables k and c are identical with those of the first set, that is: k ( t ) = k z F ∗ [ z ( t )] − [ F ∗ − F ( t )] e φt , φ = (1 − β ) [ δ + π (1 − β )] + θδβ (1 − β ) , (4) ( t ) = k z F ∗ [ z ( t )] − βσ e χt , χ = (1 − β )( δ − ρ ) + θδσ (1 − β ) , (5)where z ( t ) = [ h ( t )] − β + θ − β u ( t ) k ( t ) , F ( t ) = t Z z ( s ) σ − βσ e − ξs ds,ξ = φ − χ, F ∗ = F ∗ ( u ) = lim t →∞ F ( t ) , whereas the solutions for the variables h and u are different. In the first setthe solutions they found are: h ( t ) = h (cid:26) u e φt [ F ∗ − F ( t )] F ∗ u ( t ) (cid:27) − β − β + γ , (6) u ( t ) = ϕu ( F ∗ − F ( t ))[( ϕ + δηu ) F ∗ − δηu B ( t )] e − ϕt − δηu [ F ∗ − F ( t )] , (7)where η = 1 − β + θ − β , ϕ = ( δ + π )(1 − β ) + γδβ ,B ( t ) = t Z z ( s ) σ − βσ e − ( ξ − ϕ ) s ds, B ∗ = B ∗ ( u ) = lim t →∞ B ( t ) = (cid:18) ϕδηu (cid:19) F ∗ . The corresponding solutions determined in the second set are: h ( t ) = h (cid:26) u e φt [ F ∗ − F ( t )] F ∗ u ( t ) (cid:27) − β − β + γ , (8)but with u ( t ) given by u ( t ) = u k n z β − [ σc − ( ρ + π − πσ ) k ] + γβ (1 − σ ) k o [ F ∗ − F ( t )][ γβ (1 − σ ) − ( ρ + π − πσ ) z β − ] [ F ∗ − F ( t )] + σz β − βσ e − ξt . (9)At this point we have the following comments.1. Because the solutions for k and c are the same in both sets of solutions,we can substitute these results into the fourth equation of the system(3), to obtain: ˙ u = " ϕ − z σ − βσ e − ξt F ∗ − F ( t ) + δηu u. (10)s was proved by Chilarescu (2018 b ), the starting value u can bedetermined and is the unique solution of the equation( ϕ + δηu ) F ∗ ( u ) − δηu B ∗ ( u ) = 0 . Consequently, since the function F ( t, u ) = " ϕ − z σ − βσ e − ξt F ∗ − F ( t ) + δηu u, is continuously differentiable, than via the existence and uniquenesstheorem for nonlinear differential equations, there exists one and onlyone solution to the initial value problem ˙ u = F ( t, u ) , u = u (0) andthis solution is given by (7).2. The authors only claim that their new solution for the control variable u is an admissible solution, that is u ∈ (0 ,
1) but they provided noproof. The proof is absolutely necessary.3. If this solution really exists, then the authors would have provided theproof that this results is completely different from that one producedby Hiraguchi and Chilarescu. At least they would have been able tosupply graphs showing that the two trajectories are totally different.None of these requirements could be found in the paper of Naz andChaudhry.4 In our opinion, the so-called new solution determined by Naz andChaudhry, is nothing else, than the same solution provided by Hi-raguchi and Chilarescu, but only written in a different mathematicalformulation.
References [1] Boucekkine R. and Ruiz-Tamarit R., 2008. Special functions for thestudy of economic dynamics: The case of the Lucas-Uzawa model. Jour-nal of Mathematical Economics, 44, 33 − a . On the Solutions of the Lucas-Uzawa Model.Mathematical Modelling and Analysis. To Appear.3] Chilarescu C., 2018 b . The effect of externality on the transitional dy-namics: the case of Lucas model. Economics Bulletin, 38, 1685 − − − − −−