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Closedform solutions for the LucasUzawa 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 
In this paper, we present a dynamic picture of the two sector LucasUzawa model with logarithmic utility preferences and homogeneous technology as was proposed by Bethmann [
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, R. Perli, Uniqueness and indeterminacy: On the dynamics of endogenous growth, Journal of Economic Theory, 63 (1994), 113142. 
[3] 
D. Bethmann, Solving macroeconomic models with homogeneous technology and logarithmic preferences, Australian Economic Papers, 52 (2013), 118. 
[4] 
R. Boucekkine, J. R. RuizTamarit, Special functions for the study of economic dynamics: The case of the LucasUzawa model, Journal of Mathematical Economics, 44 (2008), 3354. doi: 10.1016/j.jmateco.2007.05.001. 
[5] 
J. Caballé, M. S. Santos, On endogenous growth with physical and human capital, Journal of Political Economy, (1993), 10421067. 
[6] 
A. Chaudhry, H. Tanveer, R. Naz, Unique and multiple equilibria in a macroeconomic model with environmental quality: An analysis of local stability, Economic Modelling, 63 (2017), 206214. 
[7] 
C. Chilarescu, On the existence and uniqueness of solution to the LucasUzawa model, Economic Modelling, 28 (2011), 109117. 
[8] 
C. Chilarescu, An analytical solutions for a model of endogenous growth, Economic Modelling, 25 (2008), 11751182. 
[9] 
C. Chilarescu, C. Sipos, Solving macroeconomic models with homogenous technology and logarithmic preferencesA note, Economics Bulletin, 34 (2014), 541550. 
[10] 
R. Lucas, On the mechanics of economic development, Journal of Monetary Economics, 22 (1988), 342. 
[11] 
O. L. Mangasarian, Sufficient conditions for the optimal control of nonlinear systems, SIAM Journal on Control, 4 (1966), 139152. doi: 10.1137/0304013. 
[12] 
C. B. Mulligan, X. SalaiMartin, Transitional dynamics in twosector models of endogenous growth, The Quaterly Journal of Economics, 108 (1993), 739773. 
[13] 
R. Naz, F. M. Mahomed, A. Chaudhry, A Partial Hamiltonian Approach for Current Value Hamiltonian Systems, Communications in Nonlinear Science and Numerical Simulation, 19 (2014), 36003610. 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), 16. 
[15] 
R. Naz, F. M. Mahomed, A. Chaudhry, A partial Lagrangian Method for Dynamical Systems, Nonlinear Dynamics, 84 (2016), 17831794. doi: 10.1007/s1107101626058. 
[16] 
R. Naz, A. Chaudhry, F. M. Mahomed, Closedform solutions for the LucasUzawa model of economic growth via the partial Hamiltonian approach, Communications in Nonlinear Science and Numerical Simulation, 30 (2016), 299306. doi: 10.1016/j.cnsns.2015.06.033. 
[17] 
R. Naz, A. Chaudhry, Comparison of closedform solutions for the LucasUzawa model via the partial Hamiltonian approach and the classical approach, Mathematical Modelling and Analysis, 22 (2017), 464483. doi: 10.3846/13926292.2017.1323035. 
[18] 
J. R. RuizTamarit, The closedform solution for a family of fourdimension nonlinear MHDS, Journal of Economic Dynamics and Control, 32 (2008), 10001014. 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), 1831. 
[20] 
D. Xie, Divergence in economic performance: Transitional dynamics with multiple equilibria, Journal of Economic Theory, 63 (1994), 97112. 
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. 
[2] 
J. Benhabib, R. Perli, Uniqueness and indeterminacy: On the dynamics of endogenous growth, Journal of Economic Theory, 63 (1994), 113142. 
[3] 
D. Bethmann, Solving macroeconomic models with homogeneous technology and logarithmic preferences, Australian Economic Papers, 52 (2013), 118. 
[4] 
R. Boucekkine, J. R. RuizTamarit, Special functions for the study of economic dynamics: The case of the LucasUzawa model, Journal of Mathematical Economics, 44 (2008), 3354. doi: 10.1016/j.jmateco.2007.05.001. 
[5] 
J. Caballé, M. S. Santos, On endogenous growth with physical and human capital, Journal of Political Economy, (1993), 10421067. 
[6] 
A. Chaudhry, H. Tanveer, R. Naz, Unique and multiple equilibria in a macroeconomic model with environmental quality: An analysis of local stability, Economic Modelling, 63 (2017), 206214. 
[7] 
C. Chilarescu, On the existence and uniqueness of solution to the LucasUzawa model, Economic Modelling, 28 (2011), 109117. 
[8] 
C. Chilarescu, An analytical solutions for a model of endogenous growth, Economic Modelling, 25 (2008), 11751182. 
[9] 
C. Chilarescu, C. Sipos, Solving macroeconomic models with homogenous technology and logarithmic preferencesA note, Economics Bulletin, 34 (2014), 541550. 
[10] 
R. Lucas, On the mechanics of economic development, Journal of Monetary Economics, 22 (1988), 342. 
[11] 
O. L. Mangasarian, Sufficient conditions for the optimal control of nonlinear systems, SIAM Journal on Control, 4 (1966), 139152. doi: 10.1137/0304013. 
[12] 
C. B. Mulligan, X. SalaiMartin, Transitional dynamics in twosector models of endogenous growth, The Quaterly Journal of Economics, 108 (1993), 739773. 
[13] 
R. Naz, F. M. Mahomed, A. Chaudhry, A Partial Hamiltonian Approach for Current Value Hamiltonian Systems, Communications in Nonlinear Science and Numerical Simulation, 19 (2014), 36003610. 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), 16. 
[15] 
R. Naz, F. M. Mahomed, A. Chaudhry, A partial Lagrangian Method for Dynamical Systems, Nonlinear Dynamics, 84 (2016), 17831794. doi: 10.1007/s1107101626058. 
[16] 
R. Naz, A. Chaudhry, F. M. Mahomed, Closedform solutions for the LucasUzawa model of economic growth via the partial Hamiltonian approach, Communications in Nonlinear Science and Numerical Simulation, 30 (2016), 299306. doi: 10.1016/j.cnsns.2015.06.033. 
[17] 
R. Naz, A. Chaudhry, Comparison of closedform solutions for the LucasUzawa model via the partial Hamiltonian approach and the classical approach, Mathematical Modelling and Analysis, 22 (2017), 464483. doi: 10.3846/13926292.2017.1323035. 
[18] 
J. R. RuizTamarit, The closedform solution for a family of fourdimension nonlinear MHDS, Journal of Economic Dynamics and Control, 32 (2008), 10001014. 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), 1831. 
[20] 
D. Xie, Divergence in economic performance: Transitional dynamics with multiple equilibria, Journal of Economic Theory, 63 (1994), 97112. 
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