On the Solutions of the Lucas-Uzawa model
Abstract
In a recent paper, Naz and Chaudry provided two solutions for the model of Lucas-Uzawa, via the Partial Hamiltonian Approach. The first one of these solutions coincides exactly with that determined by Chilarescu. For the second one, they claim that this is a new solution, fundamentally different than that obtained by Chilarescu. We will prove in this paper, using the existence and uniqueness theorem of nonlinear differential equations, that this is not at all true.
Keyword : partial Hamiltonian approach, Lucas-Uzawa model, uniqueness of solutions
How to Cite
Chilarescu, C. (2019). On the Solutions of the Lucas-Uzawa model. Mathematical Modelling and Analysis, 24(1), 127-133. https://doi.org/10.3846/mma.2019.009
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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[9] R. Naz and A. Chaudhry. Comparison of closed-form solutions for the Lucas-Uzawa model via the partial Hamiltonian approach and the classical approach. Mathematical modelling and analysis, 22(4):464–483, 2017. https://doi.org/10.3846/13926292.2017.1323035.
[10] R. Naz, F. M. Mahomed and A. Chaudhry. A partial Hamiltonian approach for current value Hamiltonian systems. Commun. Nonlinear Sci. Numer. Simulat., 19(10):3600–3610, 2014. https://doi.org/10.1016/j.cnsns.2014.03.023
[11] R. Naz, F. M. Mahomed and A. Chaudhry. Closed-form solutions for the Lucas-Uzawa model of economic growth via the partial Hamiltonian approach. Commun. Nonlinear Sci. Numer. Simulat., 30(1–3):299–306, 2016. https://doi.org/10.1016/j.cnsns.2015.06.033
[12] S. Ragni, F. Diele and C. Marangi. Steady-state invariance in highorder Runge-Kutta discretization of optimal growth models. Journal of Economic Dynamics and Control, 34(7):1248–1259, 2012. https://doi.org/10.1016/j.jedc.2010.03.006
[13] F. P. Ramsey. A mathematical theory of savings. Economic Journal, 38(152):543–559, 1928. https://doi.org/doi.org/10.2307/2224098
[14] W. T. Smith. A closed form solution to the Ramsey model. The B.E. Journal of Macroeconomics, 6(1):1–6, 2006.
[15] H. Uzawa. Optimum technical change in an aggregative model of economic growth. International Economic Review, 6(1):18–131, 1965. https://doi.org/10.2307/2525621
[16] I. Viasu. Some properties of the solution of the Ramsey model. Timisoara Journal of Economics and Business, 7(2):113–122, 2014. https://doi.org/10.1515/tjeb2015-0006
[2] R. Boucekkine and R. Ruiz-Tamarit. Special functions for the study of economic dynamics: The case of the Lucas-Uzawa model. Journal of Mathematical Economics, 44(1):33–54, 2008. https://doi.org/10.1016/j.jmateco.2007.05.001.
[3] C. Chilarescu. On the existence and uniqueness of solution to the Lucas-Uzawa model. Economic Modelling, 28(1–2):109–117, 2011. https://doi.org/10.1016/j.econmod.2010.09.019.
[4] C. Chilarescu and I. Viasu. A closed-form solution of a two-sector endogenous growth model with habit formation. Australian Economic Papers, 55(2):112– 127, 2016. https://doi.org/10.1111/1467-8454.12067.
[5] R. Hiraguchi. A note on the closed-form solution to the Lucas-Uzawa model with externality. Journal of Economic Dynamics and Control, 33(10):1757–1760, 2009. https://doi.org/10.1016/j.jedc.2009.04.001.
[6] R. Lucas. On the mechanics of economic development. Journal of Monetary Economics, 22(1):3–42, 1988. https://doi.org/10.1016/0304-3932(88)90168-7.
[7] S. Marsiglio and D. La Torre. A note on demographic shocks in a multi-sector growth model. Economics Bulletin, 32:2293–2299, 2012.
[8] S. Marsiglio and D. La Torre. Population dynamics and utilitarian criteria in the Lucas-Uzawa model. Economic Modelling, 29(4):1197–1204, 2012. https://doi.org/10.1016/j.econmod.2012.01.016
[9] R. Naz and A. Chaudhry. Comparison of closed-form solutions for the Lucas-Uzawa model via the partial Hamiltonian approach and the classical approach. Mathematical modelling and analysis, 22(4):464–483, 2017. https://doi.org/10.3846/13926292.2017.1323035.
[10] R. Naz, F. M. Mahomed and A. Chaudhry. A partial Hamiltonian approach for current value Hamiltonian systems. Commun. Nonlinear Sci. Numer. Simulat., 19(10):3600–3610, 2014. https://doi.org/10.1016/j.cnsns.2014.03.023
[11] R. Naz, F. M. Mahomed and A. Chaudhry. Closed-form solutions for the Lucas-Uzawa model of economic growth via the partial Hamiltonian approach. Commun. Nonlinear Sci. Numer. Simulat., 30(1–3):299–306, 2016. https://doi.org/10.1016/j.cnsns.2015.06.033
[12] S. Ragni, F. Diele and C. Marangi. Steady-state invariance in highorder Runge-Kutta discretization of optimal growth models. Journal of Economic Dynamics and Control, 34(7):1248–1259, 2012. https://doi.org/10.1016/j.jedc.2010.03.006
[13] F. P. Ramsey. A mathematical theory of savings. Economic Journal, 38(152):543–559, 1928. https://doi.org/doi.org/10.2307/2224098
[14] W. T. Smith. A closed form solution to the Ramsey model. The B.E. Journal of Macroeconomics, 6(1):1–6, 2006.
[15] H. Uzawa. Optimum technical change in an aggregative model of economic growth. International Economic Review, 6(1):18–131, 1965. https://doi.org/10.2307/2525621
[16] I. Viasu. Some properties of the solution of the Ramsey model. Timisoara Journal of Economics and Business, 7(2):113–122, 2014. https://doi.org/10.1515/tjeb2015-0006