American Institute of Mathematical Sciences

Advanced
August  2018, 11(4): 643-654. doi: 10.3934/dcdss.2018039

Closed-form solutions for the Lucas-Uzawa growth model with logarithmic utility preferences via the partial Hamiltonian approach

Azam Chaudhry a, and  Rehana Naz b,,

a. 

Department of Economics, Lahore School of Economics, Lahore, 53200, Pakistan
b. 

Department of Mathematics and Statistical Sciences, Lahore School of Economics, Lahore, 53200, Pakistan

* Corresponding author: Rehana Naz.

Received  November 2016 Revised  May 2017 Published  November 2017

In this paper, we present a dynamic picture of the two sector Lucas-Uzawa model with logarithmic utility preferences and homogeneous technology as was proposed by Bethmann [3] for a Robinson Crusoe economy. We use a newly developed partial Hamiltonian approach to derive a new set of closed-form solutions for the model with logarithmic utility preferences and homogeneous technology. Unlike the previous literature, our model yields three distinct closed-form solutions to the model. We establish the growth rates of all the variables which fully describe the dynamics of the model. Even though the first closed-form solution provides static growth rates and the other two provide dynamic growth rates, in the long run all the closed-form solutions approach the same static balanced growth path.

Keywords: Partial Hamiltonian approach, Lucas-Uzawa model, logarithmic utility preferences, homogeneous technology.
Mathematics Subject Classification: 37N40, 76M60, 83C15.
Citation: Azam Chaudhry, Rehana Naz. Closed-form solutions for the Lucas-Uzawa growth model with logarithmic utility preferences via the partial Hamiltonian approach. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 643-654. doi: 10.3934/dcdss.2018039
References:
[1]

K. J. Arrow, Applications of Control Theory to Economic Growth, in Veinott, A. F. , & Dantzig, G. B. (Eds. ) Mathematics of the decision sciences, Part 2, American Mathematical scociety 11,1968. Google Scholar

[2]

J. Benhabib and R. Perli, Uniqueness and indeterminacy: On the dynamics of endogenous growth, Journal of Economic Theory, 63 (1994), 113-142.   Google Scholar

[3]

D. Bethmann, Solving macroeconomic models with homogeneous technology and logarithmic preferences, Australian Economic Papers, 52 (2013), 1-18.   Google Scholar

[4]

R. Boucekkine and J. R. Ruiz-Tamarit, Special functions for the study of economic dynamics: The case of the Lucas-Uzawa model, Journal of Mathematical Economics, 44 (2008), 33-54.  doi: 10.1016/j.jmateco.2007.05.001.  Google Scholar

[5]

J. Caballé and M. S. Santos, On endogenous growth with physical and human capital, Journal of Political Economy, (1993), 1042-1067.   Google Scholar

[6]

A. Chaudhry, H. Tanveer and R. Naz, Unique and multiple equilibria in a macroeconomic model with environmental quality: An analysis of local stability, Economic Modelling, 63 (2017), 206-214.   Google Scholar

[7]

C. Chilarescu, On the existence and uniqueness of solution to the Lucas-Uzawa model, Economic Modelling, 28 (2011), 109-117.   Google Scholar

[8]

C. Chilarescu, An analytical solutions for a model of endogenous growth, Economic Modelling, 25 (2008), 1175-1182.   Google Scholar

[9]

C. Chilarescu and C. Sipos, Solving macroeconomic models with homogenous technology and logarithmic preferences-A note, Economics Bulletin, 34 (2014), 541-550.   Google Scholar

[10]

R. Lucas, On the mechanics of economic development, Journal of Monetary Economics, 22 (1988), 3-42.   Google Scholar

[11]

O. L. Mangasarian, Sufficient conditions for the optimal control of nonlinear systems, SIAM Journal on Control, 4 (1966), 139-152.  doi: 10.1137/0304013.  Google Scholar

[12]

C. B. Mulligan and X. Sala-i-Martin, Transitional dynamics in two-sector models of endogenous growth, The Quaterly Journal of Economics, 108 (1993), 739-773.   Google Scholar

[13]

R. Naz, F. M. Mahomed and A. Chaudhry, A Partial Hamiltonian Approach for Current Value Hamiltonian Systems, Communications in Nonlinear Science and Numerical Simulation, 19 (2014), 3600-3610.  doi: 10.1016/j.cnsns.2014.03.023.  Google Scholar

[14]

R. Naz, The applications of the partial Hamiltonian approach to mechanics and other areas, International Journal of Nonlinear Mechanics, 86 (2016), 1-6.   Google Scholar

[15]

R. Naz, F. M. Mahomed and A. Chaudhry, A partial Lagrangian Method for Dynamical Systems, Nonlinear Dynamics, 84 (2016), 1783-1794.  doi: 10.1007/s11071-016-2605-8.  Google Scholar

[16]

R. Naz, A. Chaudhry and F. M. Mahomed, Closed-form solutions for the Lucas-Uzawa model of economic growth via the partial Hamiltonian approach, Communications in Nonlinear Science and Numerical Simulation, 30 (2016), 299-306.  doi: 10.1016/j.cnsns.2015.06.033.  Google Scholar

[17]

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 (2017), 464-483.  doi: 10.3846/13926292.2017.1323035.  Google Scholar

[18]

J. R. Ruiz-Tamarit, The closed-form solution for a family of four-dimension nonlinear MHDS, Journal of Economic Dynamics and Control, 32 (2008), 1000-1014.  doi: 10.1016/j.jedc.2007.03.008.  Google Scholar

[19]

H. Uzawa, Optimum technical change in an aggregative model of economic growth, International Economic Review, 6 (1965), 18-31.   Google Scholar

[20]

D. Xie, Divergence in economic performance: Transitional dynamics with multiple equilibria, Journal of Economic Theory, 63 (1994), 97-112.   Google Scholar

show all references

References:
[1]

K. J. Arrow, Applications of Control Theory to Economic Growth, in Veinott, A. F. , & Dantzig, G. B. (Eds. ) Mathematics of the decision sciences, Part 2, American Mathematical scociety 11,1968. Google Scholar

[2]

J. Benhabib and R. Perli, Uniqueness and indeterminacy: On the dynamics of endogenous growth, Journal of Economic Theory, 63 (1994), 113-142.   Google Scholar

[3]

D. Bethmann, Solving macroeconomic models with homogeneous technology and logarithmic preferences, Australian Economic Papers, 52 (2013), 1-18.   Google Scholar

[4]

R. Boucekkine and J. R. Ruiz-Tamarit, Special functions for the study of economic dynamics: The case of the Lucas-Uzawa model, Journal of Mathematical Economics, 44 (2008), 33-54.  doi: 10.1016/j.jmateco.2007.05.001.  Google Scholar

[5]

J. Caballé and M. S. Santos, On endogenous growth with physical and human capital, Journal of Political Economy, (1993), 1042-1067.   Google Scholar

[6]

A. Chaudhry, H. Tanveer and R. Naz, Unique and multiple equilibria in a macroeconomic model with environmental quality: An analysis of local stability, Economic Modelling, 63 (2017), 206-214.   Google Scholar

[7]

C. Chilarescu, On the existence and uniqueness of solution to the Lucas-Uzawa model, Economic Modelling, 28 (2011), 109-117.   Google Scholar

[8]

C. Chilarescu, An analytical solutions for a model of endogenous growth, Economic Modelling, 25 (2008), 1175-1182.   Google Scholar

[9]

C. Chilarescu and C. Sipos, Solving macroeconomic models with homogenous technology and logarithmic preferences-A note, Economics Bulletin, 34 (2014), 541-550.   Google Scholar

[10]

R. Lucas, On the mechanics of economic development, Journal of Monetary Economics, 22 (1988), 3-42.   Google Scholar

[11]

O. L. Mangasarian, Sufficient conditions for the optimal control of nonlinear systems, SIAM Journal on Control, 4 (1966), 139-152.  doi: 10.1137/0304013.  Google Scholar

[12]

C. B. Mulligan and X. Sala-i-Martin, Transitional dynamics in two-sector models of endogenous growth, The Quaterly Journal of Economics, 108 (1993), 739-773.   Google Scholar

[13]

R. Naz, F. M. Mahomed and A. Chaudhry, A Partial Hamiltonian Approach for Current Value Hamiltonian Systems, Communications in Nonlinear Science and Numerical Simulation, 19 (2014), 3600-3610.  doi: 10.1016/j.cnsns.2014.03.023.  Google Scholar

[14]

R. Naz, The applications of the partial Hamiltonian approach to mechanics and other areas, International Journal of Nonlinear Mechanics, 86 (2016), 1-6.   Google Scholar

[15]

R. Naz, F. M. Mahomed and A. Chaudhry, A partial Lagrangian Method for Dynamical Systems, Nonlinear Dynamics, 84 (2016), 1783-1794.  doi: 10.1007/s11071-016-2605-8.  Google Scholar

[16]

R. Naz, A. Chaudhry and F. M. Mahomed, Closed-form solutions for the Lucas-Uzawa model of economic growth via the partial Hamiltonian approach, Communications in Nonlinear Science and Numerical Simulation, 30 (2016), 299-306.  doi: 10.1016/j.cnsns.2015.06.033.  Google Scholar

[17]

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 (2017), 464-483.  doi: 10.3846/13926292.2017.1323035.  Google Scholar

[18]

J. R. Ruiz-Tamarit, The closed-form solution for a family of four-dimension nonlinear MHDS, Journal of Economic Dynamics and Control, 32 (2008), 1000-1014.  doi: 10.1016/j.jedc.2007.03.008.  Google Scholar

[19]

H. Uzawa, Optimum technical change in an aggregative model of economic growth, International Economic Review, 6 (1965), 18-31.   Google Scholar

[20]

D. Xie, Divergence in economic performance: Transitional dynamics with multiple equilibria, Journal of Economic Theory, 63 (1994), 97-112.   Google Scholar

[1]

Rehana Naz. On sufficiency issues, first integrals and exact solutions of Uzawa-Lucas model with unskilled labor. Discrete & Continuous Dynamical Systems - S, 2020, 13 (10) : 2813-2828. doi: 10.3934/dcdss.2020170
[2]

Elvio Accinelli, Filipe Martins, Humberto Muñiz, Bruno M. P. M. Oliveira, Alberto A. Pinto. Firms, technology, training and government fiscal policies: An evolutionary approach. Discrete & Continuous Dynamical Systems - B, 2021, 26 (11) : 5723-5754. doi: 10.3934/dcdsb.2021180
[3]

Fabio Bagagiolo, Rosario Maggistro, Raffaele Pesenti. Origin-to-destination network flow with path preferences and velocity controls: A mean field game-like approach. Journal of Dynamics & Games, 2021, 8 (4) : 359-380. doi: 10.3934/jdg.2021007
[4]

Kim Dang Phung. Carleman commutator approach in logarithmic convexity for parabolic equations. Mathematical Control & Related Fields, 2018, 8 (3&4) : 899-933. doi: 10.3934/mcrf.2018040
[5]

Rehana Naz, Fazal M. Mahomed. Characterization of partial Hamiltonian operators and related first integrals. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 723-734. doi: 10.3934/dcdss.2018045
[6]

Alexander J. Zaslavski. Good programs in the RSS model without concavity of a utility function. Journal of Industrial & Management Optimization, 2006, 2 (4) : 399-423. doi: 10.3934/jimo.2006.2.399
[7]

Dong Li. A regularization-free approach to the Cahn-Hilliard equation with logarithmic potentials. Discrete & Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2021198
[8]

Jaume Llibre, Yuzhou Tian. Meromorphic integrability of the Hamiltonian systems with homogeneous potentials of degree -4. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021228
[9]

Abed Bounemoura, Edouard Pennamen. Instability for a priori unstable Hamiltonian systems: A dynamical approach. Discrete & Continuous Dynamical Systems, 2012, 32 (3) : 753-793. doi: 10.3934/dcds.2012.32.753
[10]

Claude Le Bris, Frédéric Legoll. Integrators for highly oscillatory Hamiltonian systems: An homogenization approach. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 347-373. doi: 10.3934/dcdsb.2010.13.347
[11]

Rongmei Cao, Jiangong You. The existence of integrable invariant manifolds of Hamiltonian partial differential equations. Discrete & Continuous Dynamical Systems, 2006, 16 (1) : 227-234. doi: 10.3934/dcds.2006.16.227
[12]

Yan Zhang, Peibiao Zhao, Xinghu Teng, Lei Mao. Optimal reinsurance and investment strategies for an insurer and a reinsurer under Hestons SV model: HARA utility and Legendre transform. Journal of Industrial & Management Optimization, 2021, 17 (4) : 2139-2159. doi: 10.3934/jimo.2020062
[13]

Jia Li. A malaria model with partial immunity in humans. Mathematical Biosciences & Engineering, 2008, 5 (4) : 789-801. doi: 10.3934/mbe.2008.5.789
[14]

Jérôme Coville. Nonlocal refuge model with a partial control. Discrete & Continuous Dynamical Systems, 2015, 35 (4) : 1421-1446. doi: 10.3934/dcds.2015.35.1421
[15]

Yuan Gao, Hangjie Ji, Jian-Guo Liu, Thomas P. Witelski. A vicinal surface model for epitaxial growth with logarithmic free energy. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4433-4453. doi: 10.3934/dcdsb.2018170
[16]

Zhiwu Lin. Linear instability of Vlasov-Maxwell systems revisited-A Hamiltonian approach. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021048
[17]

Guillaume Duval, Andrzej J. Maciejewski. Integrability of Hamiltonian systems with homogeneous potentials of degrees $\pm 2$. An application of higher order variational equations. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4589-4615. doi: 10.3934/dcds.2014.34.4589
[18]

Yuzhou Tian, Yulin Zhao. Global phase portraits and bifurcation diagrams for reversible equivariant Hamiltonian systems of linear plus quartic homogeneous polynomials. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2941-2956. doi: 10.3934/dcdsb.2020214
[19]

Mitsuru Shibayama. Non-integrability criterion for homogeneous Hamiltonian systems via blowing-up technique of singularities. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3707-3719. doi: 10.3934/dcds.2015.35.3707
[20]

Angelo Favini. A general approach to identification problems and applications to partial differential equations. Conference Publications, 2015, 2015 (special) : 428-435. doi: 10.3934/proc.2015.0428

2020 Impact Factor: 2.425

Tools

Metrics

  • PDF downloads (295)
  • HTML views (456)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy