Article Contents
Article Contents

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

• * Corresponding author: Rehana Naz.
• 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.

Mathematics Subject Classification: 37N40, 76M60, 83C15.

 Citation:

•  [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.