Closed-form Solutions for the Lucas-Uzawa model: Unique or Multiple
aa r X i v : . [ q -f i n . E C ] D ec Closed-form Solutions for the Lucas-Uzawamodel: Unique or Multiple
R. NazCentre for Mathematics and Statistical Sciences, Lahore School of Economics,Lahore, 53200, Pakistan ∗ Corresponding Author Email: [email protected].
Abstract
Naz and Chaudhry [3] established multiple closed-form solutions forthe basic Lucas-Uzawa model. According to Boucekkine and Ruiz-Tamarit[1] and Chilarescu [2] unique closed-form solutions exist for the basicLucas-Uzawa model. We equate expressions for variables h ( t ) and u ( t ).We provide here condition for the unique closed-form solution and pro-posed an open question for evaluation of integral in closed-form. A similaranalysis is carried out for the Lucas-Uzawa model with logarithmic utilitypreferences. The following model is discussed for the closed form solutions by Boucekkineand Ruiz-Tamarit [1], Chilarescu [2] and Naz and Chaudhry [3] for fairly generalvalues of parameters. The representative agent’s utility function is defined as
M ax c,u Z ∞ c − σ − − σ e − ρt , σ = 1 (1)subject to the constraints of physical capital and human capital:˙ k ( t ) = γk β u − β h − β − πk − c, k = k (0)˙ h ( t ) = δ (1 − u ) h, h = h (0) . (2)Recently, Bethmann [5] developed a stylized version of the two sector Lucas-Uzawa model with logarithmic utility preferences and solved the model by dy-namic programming technique. Chilarescu and Sipos [6] derived closed-formsolutions for the variables in the model proposed by Bethmann in terms of nu-merically computable functions involving integrals. Chaudhry and Naz [7] de-rived multiple closed-form solutions for this model. The representative agent’sutility function is defined as M ax c,u Z ∞ e − ρt ln( c ) dt, (3)subject to the constraints of physical capital and human capital:˙ k ( t ) = Ak α ( uh ) − α − c, k = k (0) , ˙ h ( t ) = δ (1 − u ) h, h = h (0) , (4)1here ρ > α is the elasticity of output with respect tophysical capital, A > δ > k is physical capital, h is humancapital, c is per capita consumption and u is the fraction of labor allocated tothe production of physical capital. The following closed-form solution derived via two first integrals I and I isgiven in equation (3.21) on page 474 of Naz and Chaudhry [3]: c ( t ) = c z βσ e − ( ρ − δ ) σ t z − βσ ,k ( t ) = (cid:18) k c z β − σσ − F ( t ) (cid:19) c z βσ z ( t ) − e ( δ + π − πβ ) β t ,h ( t ) = h z [ σc z β − − ( ρ + π − πσ ) k z β − + βγ (1 − σ ) k ] [ σc z βσ e − ( ρ − δ ) σ t z − βσ + β +( βγ (1 − σ ) − ( ρ + π − πσ ) z β − )( k c z β − σσ − F ( t )) c z βσ e ( δ + π − πβ ) β t ] ,u ( t ) = u k [ σc z β − − ( ρ + π − πσ ) k z β − + βγ (1 − σ ) k ] × ( k c z β − σσ − F ( t ))[ βγ (1 − σ ) − ( ρ + π − πσ ) z β − ]( k c z β − σσ − F ( t )) + σz β − βσ e − ( δ + π − πββ − δ − ρσ ) t ,λ ( t ) = c − σ z − β e ( ρ − δ ) t z β ,µ ( t ) = c e ( ρ − δ ) t , F ( t ) = Z t z ( t ) σ − βσ e − ( δ + π − πββ − δ − ρσ ) t dt,z ( t ) = z ∗ z [( z ∗ − β − z − β ) e − (1 − β )( δ + π ) β t + z − β ] − β , (5)lim t →∞ F ( t ) = k c z β − σσ ,ρ < δ < ρ + δσ, δ + π − πββ − δ − ρσ > ,c z βσ = (cid:18) c δ (1 − β ) γ (cid:19) − σ ,γ (1 − β )( ρ − δ + δσ ) δ = u k [ σc z β − − ( ρ + π − πσ ) k z β − + βγ (1 − σ ) k ] ,z ∗ = (cid:18) βγδ + π (cid:19) β − . The following closed-form solution via one first integral I is given in equation(4.6) on page 476 of Naz and Chaudhry [3]: c ( t ) = c z βσ e − ( ρ − δ ) σ t z − βσ ,k ( t ) = (cid:18) k c z β − σσ − F ( t ) (cid:19) c z βσ z ( t ) − e ( π + δ − πβ ) β t ,h ( t ) = (cid:20)(cid:18) ( δ + π )(1 − β ) β k c z β − σσ + δu k c z β − σσ − δu G ( t ) (cid:19) e − ( δ + π )(1 − β ) β t − δu ( k c z β − σσ − F ( t )) (cid:21) × c z βσ δ + π )(1 − β ) β u e ( π + δ − πβ ) β t ,u ( t ) = ( δ + π )(1 − β ) β u [ k c z β − σσ − F ( t )][( ( δ + π )(1 − β ) β + δu ) k c z β − σσ − δu G ( t )] e − ( δ + π )(1 − β ) β t − δu [ k c z β − σσ − F ( t )] λ ( t ) = c − σ z − β e ( ρ − δ ) t z β ,µ ( t ) = c e ( ρ − δ ) t , where 3 < δ < ρ + δσ, δ + π − πββ − δ − ρσ > ,F ( t ) = Z t z ( t ) σ − βσ e − ( δ + π − πββ − δ − ρσ ) t dt,G ( t ) = Z t z ( t ) σ − βσ e − δσ − δ + ρσ t dt, (6) z ( t ) = z ∗ z [( z ∗ − β − z − β ) e − (1 − β )( δ + π ) β t + z − β ] − β ,c z βσ = (cid:18) c δ (1 − β ) γ (cid:19) − σ , lim t →∞ F ( t ) = k c z β − σσ , lim t →∞ (cid:20) ( ( δ + π )(1 − β ) β + δu ) k c z β − σσ − δu G ( t ) (cid:21) = 0 , lim t →∞ G ( t ) = ( ( δ + π )(1 − β ) β + δu ) δu lim t →∞ F ( t ) ,z ∗ = (cid:18) βγδ + π (cid:19) β − . Chilarescu [2] derived same solution given on page 113 in Theorem 1 by classi-cal approach and utilized numerical simulations to evaluate functions F ( t ) and G ( t ). Boucekkine and Ruiz-Tamarit [1] derived a similar solution and they ex-pressed unknown functions similar to F ( t ) and G ( t ) in terms of Hypergeometricfunctions. Naz and Chaudhry [3] claimed that in closed-form solutions (5) and(6) the expressions for the variables c ( t ), k ( t ) are same but expressions for thevariables h ( t ) and u ( t ) are different. Thus closed-form solution (5) is differentfrom closed-form solution (6).The uniqueness of solution discussed by Boucekkine and Ruiz-Tamarit [1],Chilarescu [2] indicates that the expressions for variables h ( t ) and u ( t ) in closed-form (5) and (6) should be same. We equate expression for h ( t ) and u ( t ) in(5) and (6), after simplifications, we obtain following expression for unknownfunction G ( t ) in terms of F ( t ): G ( t ) = F ∗ − ( F ∗ − F ( t )) e ( δ + π )(1 − β ) β t + ( δ + π )(1 − β ) β F ∗ δu − e ( δ + π )(1 − β ) β t ( δ + π )(1 − β ) β γ (1 − β )[ ρ − δ (1 − σ )] × (cid:20) σz β − βσ e − ( δ + π (1 − β ) β − δ − ρσ ) t + (cid:18) γβ (1 − σ ) − (cid:0) ρ + π − πσ (cid:1) z ( t ) β − (cid:19) ( F ∗ − F ( t )) (cid:21) , (7)4rovided following condition holds γ (1 − β )( ρ − δ + δσ ) δ = u k [ σc z β − − ( ρ + π − πσ ) k z β − + βγ (1 − σ ) k ] , (8)where F ∗ = k c z β − σσ . It is important to mention here that condition (8) arisessystematically for the closed-form solution (5).In (6) the expression for G ( t ) is G ( t ) = Z t z ( s ) σ − βσ e − ζs ds, ζ = (cid:18) δ + π (1 − β ) β − δ − ρσ (cid:19) − ( δ + π )(1 − β ) β (9)From (7) and (9), we deduce that Z t z ( s ) σ − βσ e − ζs ds = F ∗ − ( F ∗ − F ( t )) e ( δ + π )(1 − β ) β t + ( δ + π )(1 − β ) β F ∗ δu − e ( δ + π )(1 − β ) β t ( δ + π )(1 − β ) β γ (1 − β )[ ρ − δ (1 − σ )] × (cid:20) σz β − βσ e − ( δ + π (1 − β ) β − δ − ρσ ) t + (cid:18) γβ (1 − σ ) − (cid:0) ρ + π − πσ (cid:1) z ( t ) β − (cid:19) ( F ∗ − F ( t )) (cid:21) , (10)provided condition (8) holds. If one can proof (10) as true only for that casethe expressions for the variables h ( t ) and u ( t ) in closed-form (5) and (6) will besame. Thus (5) and (6) provided by Naz and Chaudhry [3] takes same form.This is consistent with Chilarescu [2] and Boucekkine and Ruiz-Tamarit [1].If G ( t ) is different from (10) then multiple closed-form solutions exist forthe Lucas-Uzawa model for fairly general values of parameters. It is an openquestion to prove (10) in closed-form and not numerically. σ = β ( ρ + π )2 πβ − δ + δβ − π Naz et al [4] provided a closed-form solution under a specific parametric restric-tion σ = β ( ρ + π )2 πβ − δ + δβ − π provided 2 πβ − δ + δβ − π > σ >
0. Theparametric restriction arises automatically and it was important to mention thissolution which at the moment seems purely mathematical solution. It might beinteresting for economists to test it empirically and it is an open question totest this empirically. 5
Closed-form solutions for Lucas-Uzawa modelwith logarithmic utility preferences: Uniqueor multiple