# American Institute of Mathematical Sciences

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

 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.

Citation: Azam Chaudhry, Rehana Naz. Closed-form solutions for the Lucas-Uzawa growth model with logarithmic utility preferences via the partial Hamiltonian approach. Discrete and 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. [2] J. Benhabib and R. Perli, Uniqueness and indeterminacy: On the dynamics of endogenous growth, Journal of Economic Theory, 63 (1994), 113-142. [3] D. Bethmann, Solving macroeconomic models with homogeneous technology and logarithmic preferences, Australian Economic Papers, 52 (2013), 1-18. [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. [5] J. Caballé and M. S. Santos, On endogenous growth with physical and human capital, Journal of Political Economy, (1993), 1042-1067. [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. [7] C. Chilarescu, On the existence and uniqueness of solution to the Lucas-Uzawa model, Economic Modelling, 28 (2011), 109-117. [8] C. Chilarescu, An analytical solutions for a model of endogenous growth, Economic Modelling, 25 (2008), 1175-1182. [9] C. Chilarescu and C. Sipos, Solving macroeconomic models with homogenous technology and logarithmic preferences-A note, Economics Bulletin, 34 (2014), 541-550. [10] R. Lucas, On the mechanics of economic development, Journal of Monetary Economics, 22 (1988), 3-42. [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. [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. [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. [14] R. Naz, The applications of the partial Hamiltonian approach to mechanics and other areas, International Journal of Nonlinear Mechanics, 86 (2016), 1-6. [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. [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. [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. [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. [19] H. Uzawa, Optimum technical change in an aggregative model of economic growth, International Economic Review, 6 (1965), 18-31. [20] D. Xie, Divergence in economic performance: Transitional dynamics with multiple equilibria, Journal of Economic Theory, 63 (1994), 97-112.

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##### 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. [2] J. Benhabib and R. Perli, Uniqueness and indeterminacy: On the dynamics of endogenous growth, Journal of Economic Theory, 63 (1994), 113-142. [3] D. Bethmann, Solving macroeconomic models with homogeneous technology and logarithmic preferences, Australian Economic Papers, 52 (2013), 1-18. [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. [5] J. Caballé and M. S. Santos, On endogenous growth with physical and human capital, Journal of Political Economy, (1993), 1042-1067. [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. [7] C. Chilarescu, On the existence and uniqueness of solution to the Lucas-Uzawa model, Economic Modelling, 28 (2011), 109-117. [8] C. Chilarescu, An analytical solutions for a model of endogenous growth, Economic Modelling, 25 (2008), 1175-1182. [9] C. Chilarescu and C. Sipos, Solving macroeconomic models with homogenous technology and logarithmic preferences-A note, Economics Bulletin, 34 (2014), 541-550. [10] R. Lucas, On the mechanics of economic development, Journal of Monetary Economics, 22 (1988), 3-42. [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. [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. [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. [14] R. Naz, The applications of the partial Hamiltonian approach to mechanics and other areas, International Journal of Nonlinear Mechanics, 86 (2016), 1-6. [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. [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. [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. [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. [19] H. Uzawa, Optimum technical change in an aggregative model of economic growth, International Economic Review, 6 (1965), 18-31. [20] D. Xie, Divergence in economic performance: Transitional dynamics with multiple equilibria, Journal of Economic Theory, 63 (1994), 97-112.
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