# American Institute of Mathematical Sciences

October  2020, 13(10): 2813-2828. doi: 10.3934/dcdss.2020170

## On sufficiency issues, first integrals and exact solutions of Uzawa-Lucas model with unskilled labor

 Centre for Mathematics and Statistical Sciences, Lahore School of Economics, Lahore, 53200, Pakistan

* Corresponding author: Rehana Naz

Received  January 2019 Revised  May 2019 Published  October 2020 Early access  December 2019

In this article, the sufficiency issues, first integrals and exact solutions for the Uzawa-Lucas model with unskilled labor are investigated. The sufficient conditions are established by utilizing Arrow's Sufficiency theorem. The non-negativeness conditions for the balanced growth path (BGP) are provided and growth rate is explicitly given in terms of parameters of the model. The first integrals are established by the partial Hamiltonian approach. Then first integrals are utilized to construct the exact solutions for all the variables. The growth rates of all variables and graphical representation of exact solutions are provided for the special case when the inverse of the intertemporal elasticity of substitution is the same as the share of physical capital.

Citation: Rehana Naz. On sufficiency issues, first integrals and exact solutions of Uzawa-Lucas model with unskilled labor. Discrete and Continuous Dynamical Systems - S, 2020, 13 (10) : 2813-2828. doi: 10.3934/dcdss.2020170
##### References:
 [1] K. J. Arrow, Applications of control theory to economic growth, Mathematics of the Decision Sciences, Part 2, American Mathematical Society, Providence, R.I., (1968), 85–119. [2] J. Caballé and M. S. Santos, On endogenous growth with physical and human capital, Journal of Political Economy, (1993), 1042–1067. [3] V. V. Chari, L. E. Jones and R. E. Manuelli, The growth effects of monetary policy, Federal Reserve Bank of Minneapolis, Quarterly Review-Federal Reserve Bank of Minneapolis, 19 (1995), 18. [4] A. Chaudhry and R. Naz, Closed-form solutions for the Lucas-Uzawa Growth model with logarithmic utility preferences via the partial Hamiltonian approach, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), 643-654.  doi: 10.3934/dcdss.2018039. [5] A. F. Cheviakov and R. Naz, A recursion formula for the construction of local conservation laws of differential equations, Journal of Mathematical Analysis and Applications, 448 (2017), 198-212.  doi: 10.1016/j.jmaa.2016.10.042. [6] A. C. Chiang, Elements of Dynamic Optimization, Illinois: Waveland Press Inc., 2000. [7] R. P. Cysne, A note on the non-convexity problem in some shopping-time and human-capital models, Journal of Banking & Finance, 30 (2006), 2737-2745. [8] R. P. Cysne, A note on Inflation and Welfare, Journal of Banking & Finance, 32 (2008), 1984-1987. [9] M. Ferrara and L. Guerrini, A note on the Uzawa-Lucas model with unskilled labor, Applied Sciences, 12 (2010), 90-95. [10] B. U. Haq and I. Naeem, First integrals and analytical solutions of some dynamical systems, Nonlinear Dynamics, (2018), 1–19. [11] M. I. Kamien and N. L. Schwartz, Sufficient conditions in optimal control theory, Journal of Economic Theory, 3 (1971), 207-214.  doi: 10.1016/0022-0531(71)90018-4. [12] A. H. Kara, F. M. Mahomed, I. Naeem and C. Wafo Soh, Partial Noether operators and first integrals via partial Lagrangians, Mathematical Methods in the Applied Sciences, 30 (2007), 2079-2089.  doi: 10.1002/mma.939. [13] R. E. Lucas Jr., Inflation and welfare, Monetary Theory as a Basis for Monetary Policy, Palgrave Macmillan UK, (2001), 96–142. [14] R. E. Lucas Jr., On the mechanics of economic development, Journal of Monetary Economics, 22 (1988), 3-42. [15] W.-X. Ma, Conservation laws by symmetries and adjoint symmetries, Discr. Cont. Dyn. Sys. Ser. S, 11 (2018), 707-721.  doi: 10.3934/dcdss.2018044. [16] W.-X. Ma and M. Chen, Hamiltonian and quasi-Hamiltonian structures associated with semi-direct sums of Lie algebras, Journal of Physics A: Mathematical and General, 39 (2006), 10787-10801.  doi: 10.1088/0305-4470/39/34/013. [17] K. S. Mahomed and R. J. Moitsheki, First integrals of generalized Ermakov systems via the Hamiltonian formulation, International Journal of Modern Physics-B, 30 (2016), 1640019, 12 pp. doi: 10.1142/S0217979216400191. [18] O. L. Mangasarian, Sufficient conditions for the optimal control of nonlinear systems, SIAM Journal on Control, 4 (1966), 139-152.  doi: 10.1137/0304013. [19] C. B. Mulligan and X. Sala-i-Martin, Transitional Dynamics in Two-Sector Models of Endogenous Growth (No. w3986), National Bureau of Economic Research, 1992. [20] 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. [21] R. Naz, The applications of the partial Hamiltonian approach to mechanics and other areas, International Journal of Non-Linear Mechanics, 86 (2016), 1-6. [22] 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. [23] R. Naz, A. Chaudhry and F. M. Mahomed, Closed-form solutions for the LucasUzawa 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. [24] 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. [25] R. Naz and A. Chaudhry, Closed-form solutions of Lucas-Uzawa model with externalities via partial Hamiltonian approach, Computational and Applied Mathematics, 37 (2018), 5146-5161.  doi: 10.1007/s40314-018-0622-6. [26] R. Naz and I. Naeem, The artificial hamiltonian, first integrals, and closed-form solutions of dynamical systems for epidemics, Zeitschrift fr Naturforschung A, 73 (2018), 323-330. [27] R. Naz, F. M. Mahomed and D. P. Mason, Comparison of different approaches to conservation laws for some partial differential equations in fluid mechanics, Applied Mathematics and Computation, 205 (2008), 212-230.  doi: 10.1016/j.amc.2008.06.042. [28] R. Naz, I. L. Freire and I. Naeem, Comparison of different approaches to construct first integrals for ordinary differential equations, Abstr. Appl. Anal., (2014), Art. ID 978636, 15 pp. doi: 10.1155/2014/978636. [29] L. S. Pontryagin, Mathematical Theory of Optimal Processes, CRC Press, 1987. [30] P. E. Robertson, Demographic shocks and human capital accumulation in the Uzawa-Lucas model, Economics Letter, 74 (2002), 151-156.  doi: 10.1016/S0165-1765(01)00547-X. [31] H. Uzawa, Optimum technical change in an aggregative model of economic growth, International Economic Review, 6 (1965), 18-31. [32] T. Wolf, A comparison of four approaches to the calculation of conservation laws, European Journal of Applied Mathematics, 13 (2002), 129-152.  doi: 10.1017/S0956792501004715.

show all references

##### References:
 [1] K. J. Arrow, Applications of control theory to economic growth, Mathematics of the Decision Sciences, Part 2, American Mathematical Society, Providence, R.I., (1968), 85–119. [2] J. Caballé and M. S. Santos, On endogenous growth with physical and human capital, Journal of Political Economy, (1993), 1042–1067. [3] V. V. Chari, L. E. Jones and R. E. Manuelli, The growth effects of monetary policy, Federal Reserve Bank of Minneapolis, Quarterly Review-Federal Reserve Bank of Minneapolis, 19 (1995), 18. [4] A. Chaudhry and R. Naz, Closed-form solutions for the Lucas-Uzawa Growth model with logarithmic utility preferences via the partial Hamiltonian approach, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), 643-654.  doi: 10.3934/dcdss.2018039. [5] A. F. Cheviakov and R. Naz, A recursion formula for the construction of local conservation laws of differential equations, Journal of Mathematical Analysis and Applications, 448 (2017), 198-212.  doi: 10.1016/j.jmaa.2016.10.042. [6] A. C. Chiang, Elements of Dynamic Optimization, Illinois: Waveland Press Inc., 2000. [7] R. P. Cysne, A note on the non-convexity problem in some shopping-time and human-capital models, Journal of Banking & Finance, 30 (2006), 2737-2745. [8] R. P. Cysne, A note on Inflation and Welfare, Journal of Banking & Finance, 32 (2008), 1984-1987. [9] M. Ferrara and L. Guerrini, A note on the Uzawa-Lucas model with unskilled labor, Applied Sciences, 12 (2010), 90-95. [10] B. U. Haq and I. Naeem, First integrals and analytical solutions of some dynamical systems, Nonlinear Dynamics, (2018), 1–19. [11] M. I. Kamien and N. L. Schwartz, Sufficient conditions in optimal control theory, Journal of Economic Theory, 3 (1971), 207-214.  doi: 10.1016/0022-0531(71)90018-4. [12] A. H. Kara, F. M. Mahomed, I. Naeem and C. Wafo Soh, Partial Noether operators and first integrals via partial Lagrangians, Mathematical Methods in the Applied Sciences, 30 (2007), 2079-2089.  doi: 10.1002/mma.939. [13] R. E. Lucas Jr., Inflation and welfare, Monetary Theory as a Basis for Monetary Policy, Palgrave Macmillan UK, (2001), 96–142. [14] R. E. Lucas Jr., On the mechanics of economic development, Journal of Monetary Economics, 22 (1988), 3-42. [15] W.-X. Ma, Conservation laws by symmetries and adjoint symmetries, Discr. Cont. Dyn. Sys. Ser. S, 11 (2018), 707-721.  doi: 10.3934/dcdss.2018044. [16] W.-X. Ma and M. Chen, Hamiltonian and quasi-Hamiltonian structures associated with semi-direct sums of Lie algebras, Journal of Physics A: Mathematical and General, 39 (2006), 10787-10801.  doi: 10.1088/0305-4470/39/34/013. [17] K. S. Mahomed and R. J. Moitsheki, First integrals of generalized Ermakov systems via the Hamiltonian formulation, International Journal of Modern Physics-B, 30 (2016), 1640019, 12 pp. doi: 10.1142/S0217979216400191. [18] O. L. Mangasarian, Sufficient conditions for the optimal control of nonlinear systems, SIAM Journal on Control, 4 (1966), 139-152.  doi: 10.1137/0304013. [19] C. B. Mulligan and X. Sala-i-Martin, Transitional Dynamics in Two-Sector Models of Endogenous Growth (No. w3986), National Bureau of Economic Research, 1992. [20] 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. [21] R. Naz, The applications of the partial Hamiltonian approach to mechanics and other areas, International Journal of Non-Linear Mechanics, 86 (2016), 1-6. [22] 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. [23] R. Naz, A. Chaudhry and F. M. Mahomed, Closed-form solutions for the LucasUzawa 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. [24] 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. [25] R. Naz and A. Chaudhry, Closed-form solutions of Lucas-Uzawa model with externalities via partial Hamiltonian approach, Computational and Applied Mathematics, 37 (2018), 5146-5161.  doi: 10.1007/s40314-018-0622-6. [26] R. Naz and I. Naeem, The artificial hamiltonian, first integrals, and closed-form solutions of dynamical systems for epidemics, Zeitschrift fr Naturforschung A, 73 (2018), 323-330. [27] R. Naz, F. M. Mahomed and D. P. Mason, Comparison of different approaches to conservation laws for some partial differential equations in fluid mechanics, Applied Mathematics and Computation, 205 (2008), 212-230.  doi: 10.1016/j.amc.2008.06.042. [28] R. Naz, I. L. Freire and I. Naeem, Comparison of different approaches to construct first integrals for ordinary differential equations, Abstr. Appl. Anal., (2014), Art. ID 978636, 15 pp. doi: 10.1155/2014/978636. [29] L. S. Pontryagin, Mathematical Theory of Optimal Processes, CRC Press, 1987. [30] P. E. Robertson, Demographic shocks and human capital accumulation in the Uzawa-Lucas model, Economics Letter, 74 (2002), 151-156.  doi: 10.1016/S0165-1765(01)00547-X. [31] H. Uzawa, Optimum technical change in an aggregative model of economic growth, International Economic Review, 6 (1965), 18-31. [32] T. Wolf, A comparison of four approaches to the calculation of conservation laws, European Journal of Applied Mathematics, 13 (2002), 129-152.  doi: 10.1017/S0956792501004715.
Evolution over time of $c(t), \;u(t), \; k(t)$ and $h(t)$
Effect of change of $\beta$ on evolution over time of $c(t), \;u(t), \; k(t)$ and $h(t)$
Parameters values
 $\theta=\alpha$ $\beta$ $\rho$ $\nu$ $n$ $m$ $k_0$ $h_0$ $0.33$ $0.25$ $0.05$ $0.1$ $0.04$ $0.01$ $40$ $10$
 $\theta=\alpha$ $\beta$ $\rho$ $\nu$ $n$ $m$ $k_0$ $h_0$ $0.33$ $0.25$ $0.05$ $0.1$ $0.04$ $0.01$ $40$ $10$
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