aa r X i v : . [ ec on . T H ] J u l On the Solutions of the Lucas-Uzawa Model
Constantin Chilarescu
Laboratoire CLERSE Universit´e de Lille, FranceE-mail: [email protected]
Abstract
In a recent paper, Naz and Chaudry provided two solutions for themodel of Lucas-Uzawa, via the Partial Hamiltonian Approach. Thefirst one of these solutions coincides exactly with that determined byChilarescu. For the second one, they claim that this is a new solution,fundamentally different than that obtained by Chilarescu. We willprove in this paper, using the existence and uniqueness theorem ofnonlinear differential equations, that this is not at all true.
Keywords : Partial Hamiltonian approach, Lucas-Uzawa model, Uniquenessof solutions.
AMS Subject Classification : 35L65; 76M60; 83C15
The model of Lucas-Uzawa is characterized by the well-known optimizationproblem.
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 ) = γk ( t ) β [ u ( t ) h ( t )] − β − πk ( t ) − c ( t ) , ˙ h ( t ) = δ [1 − u ( t )] h ( t ) ,k = k (0) , h = h (0) , (2) where k > and h > are given, β is the elasticity of output with respectto physical capital, ρ is a positive discount factor, the efficiency parameters γ > and δ > represent the constant technological levels in the good sectorand, respectively in the education sector, k is physical capital, h is humancapital, c is the real per-capita consumption and u is the fraction of laborallocated to the production of physical capital. σ − represents the constantelasticity of intertemporal substitution, and throughout this paper we supposethat σ = β . The dynamical system that drives the economy over time is given by ˙ kk = γ (cid:0) huk (cid:1) − β − π − ck ˙ hh = δ (1 − u ) ˙ λλ = ρ + π − βγ (cid:0) huk (cid:1) − β ˙ µµ = ρ − δ ˙ cc = − ρ + πσ + γβσ (cid:0) huk (cid:1) − β ˙ uu = ϕ − ck + δu, ϕ = ( δ + π )(1 − β ) β . (3)In two recent papers Naz et al. (2014 , et al. (2012), Viasu (2014), Hiraguchi (2009)and, Chilarescu and Viasu (2016).Naz and Chaudry obtained three first integrals, denoted by I , I and I , the first two with no restrictions on parameters and the last one witha restriction on parameters. Among the two first integrals, only I enablesus to obtain directly the solutions for the Lucas model. It is impossible toobtain solutions for the Lucas model by using only the second integral I .That is why it is necessary to combine the two first integrals I and I inorder to obtain the solutions. The solutions thus obtained for the variables k and c , coincide exactly with those of the previous case, but the solutionsfor the variables h and u do not coincide with those of the previous case.If the solution for u is really a new solution, then the authors have toprove that it is an admissible solution, i.e. 0 < u < u and c , becausethese initial values are unknown. None of these requirements can be foundin the papers of the cited authors. In the next section we will prove that theLucas-Uzawa model admits a unique solution and thus the claim of Naz andChaudry on the existence of multiple solutions is inexact. In order to solve the system (3), Chilarescu introduced the new variable z = huk and thus he obtained the following differential equation˙ z = (cid:20) δ + πβ − γz − β (cid:21) z, whose solution is given by z ( t ) = z − β ∗ z − β (cid:16) z − β ∗ − z − β (cid:17) e − ϕt + z − β − β . (4)3s was proved by Chilarescu and by Naz and Chaudry, the solutions for k and c of the system (3) are given by c ( t ) = h u A ∗ [ z ( t )] − βσ e δ − ρσ t , (5) k ( t ) = h u A ∗ [ A ∗ − A ( t )] [ z ( t )] − e φt , φ = δ + π (1 − β ) β , (6) c ( t ) k ( t ) = [ z ( t )] σ − βσ e − ξt A ∗ − A ( t ) , ξ = φ − δ − ρσ , (7)where A ( t ) = t Z z ( s ) σ − βσ e − ξs ds, A ∗ = lim t →∞ A ( t ) . Substituting (7) into the last equation of the system (3) we arrive at thefollowing nonlinear differential equation˙ u = " ϕ − z σ − βσ e − ξt A ∗ − A ( t ) + δu u. As was proved by Chilarescu, the starting value u can be determined and isthe unique solution of the equation( ϕ + δu ) A ∗ ( u ; k , h ) − δu B ∗ ( u ; k , h ) = 0 . Consequently, since the function F ( t, u ) = " ϕ − z σ − βσ e − ξt A ∗ − A ( t ) + δu u, is continuously differentiable, than via the existence and uniqueness theoremfor nonlinear differential equations, there exists one and only one solution tothe initial value problem ˙ u = F ( t, u ) , u = u (0) . This solution is given by u ( t ) = ϕu [ A ∗ − A ( t )][( ϕ + δu ) A ∗ − δu B ( t )] e − ϕt − δu [ A ∗ − A ( t )] , (8)4here B ( t ) = t Z z ( s ) σ − βσ e − ( ξ − ϕ ) s ds, B ∗ = lim t →∞ B ( t ) . Therefore, the claim of Naz et al. and Naz and Chaudry concerning theexistence of multiple solutions for the Lucas-Uzawa model is inexact.In fact, the second set of solutions determined by Naz and Chaudry isidentical to that in the first case, but is only written in a different mathemat-ical formulation. The solutions for the state variable k and for the controlvariable c coincide exactly to those determined in the first set of solutions(the same solutions were determined by Chilarescu). Only the solutions forthe state variable h and for the control variable u were determined in a differ-ent mathematical formulation. These solutions, written in accordance withthe notations used in this paper, are: h ( t ) = n [ A ∗ − A ( t )] (cid:2) γβ (1 − σ ) − ( ρ + π − πσ ) z ( t ) β − (cid:3) + σz ( t ) β − βσ e − ξt o e φt δ − γ (1 − β )( ρ − δ + δσ ) c − z − βσ , (9) u ( t ) = δ − γ (1 − β )( ρ − δ + δσ ) [ A ∗ − A ( t )][ A ∗ − A ( t )] [ γβ (1 − σ ) − ( ρ + π − πσ ) z ( t ) β − ] + σz ( t ) β − βσ e − ξt . (10)If in the paper of Chilarescu, we express the function B in terms of thefunction A , i.e., B ( t ) = ϕ + δu δu A ∗ − [ A ∗ − A ( t )] (cid:2) µ − χz ( t ) β − (cid:3) e ϕt − ωz ( t ) β − βσ e − ( ξ − ϕ ) t , with µ = γβϕ (1 − σ ) η , χ = ϕ [ ρ + π (1 − σ ] η , ω = σϕη , η = γ (1 − β ) [ ρ − δ (1 − σ )] . and then substitute this result into the corresponding equations of u and h given in the paper of Chilarescu, we obtain exactly the same results as thoseof the equations obtained by Naz and Chaudry.The cited authors also claim that under the specific restriction σ = ( ρ + π ) βπβ − ( δ + π )(1 − β ) = 1 πρ + π − δ + πρ + π − ββ , (11)5here exists another solution of the Lucas-Uzawa model. The problem hereis that we cannot choose arbitrarily the values of all parameters in the modelof Lucas. As it is well-known, β represents the capital share of income. Forexample, if we choose π = 0 . δ = 0 . ρ = 0 .
04 which are acceptablevalues, then we have to choose (for example) β = 0 .
8, value that generatesfor σ = 4. These two values, for β and σ are certainly beyond the values con-firmed by the econometric estimations and consequently this second solutioncould be considered only as a purely mathematical alternative.In the next section we present some numerical simulations in order toshow that the trajectories determined by Naz et al. or by Naz and Chaudry(equations (9) and (10)), coincide exactly with those provided by Chilarescu. Conclusions and some numerical simulations
The uniqueness of the solutions of Lucas’s model was proved for the firsttime by Boucekkine and Ruiz-Tamarit and later by Chilarescu, by using com-pletely different mathematical techniques. Naz et al. and Naz and Chaudryrecently published several papers in which they claim that Lucas’s model,without any restrictions on the parameters, presents multiple solutions. Ob-viously this claim is not true and this paper clarifies definitively this subject.We proved our result, via the theorem of existence and uniqueness of the non-linear differential equations. Examining the results presented in the papersby these authors, results obtained via the partial Hamiltonian approach, weconclude that this method could not provide new general results, but onlyto confirm the old results obtained by other papers. What is really newto this method is the fact that it can produce some particular solutions,obtained by using different restrictions on the parameters. In order to givemore credibility to the results obtained in this paper, we present below, somenumerical simulations. To do this, we consider here a well-known benchmarkeconomy: β = 0 . , γ = 1 . , δ = 0 . , π = 0 . , ρ = 0 . , σ = 1 . , h = 10 , and k = 80 and the results are presented into the four graphs, denoted by F ig.N o. − F ig.N o.