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  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 & 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.  Google Scholar

[2]

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

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

A. C. Chiang, Elements of Dynamic Optimization, Illinois: Waveland Press Inc., 2000. Google Scholar

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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.   Google Scholar

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R. P. Cysne, A note on Inflation and Welfare, Journal of Banking & Finance, 32 (2008), 1984-1987.   Google Scholar

[9]

M. Ferrara and L. Guerrini, A note on the Uzawa-Lucas model with unskilled labor, Applied Sciences, 12 (2010), 90-95.   Google Scholar

[10]

B. U. Haq and I. Naeem, First integrals and analytical solutions of some dynamical systems, Nonlinear Dynamics, (2018), 1–19. Google Scholar

[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.  Google Scholar

[12]

A. H. KaraF. M. MahomedI. 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.  Google Scholar

[13]

R. E. Lucas Jr., Inflation and welfare, Monetary Theory as a Basis for Monetary Policy, Palgrave Macmillan UK, (2001), 96–142. Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[20]

R. NazF. 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

[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.   Google Scholar

[22]

R. NazF. 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

[23]

R. NazA. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[27]

R. NazF. 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.  Google Scholar

[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.  Google Scholar

[29] L. S. Pontryagin, Mathematical Theory of Optimal Processes, CRC Press, 1987.   Google Scholar
[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.  Google Scholar

[31]

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

[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.  Google Scholar

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.  Google Scholar

[2]

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

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

A. C. Chiang, Elements of Dynamic Optimization, Illinois: Waveland Press Inc., 2000. Google Scholar

[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.   Google Scholar

[8]

R. P. Cysne, A note on Inflation and Welfare, Journal of Banking & Finance, 32 (2008), 1984-1987.   Google Scholar

[9]

M. Ferrara and L. Guerrini, A note on the Uzawa-Lucas model with unskilled labor, Applied Sciences, 12 (2010), 90-95.   Google Scholar

[10]

B. U. Haq and I. Naeem, First integrals and analytical solutions of some dynamical systems, Nonlinear Dynamics, (2018), 1–19. Google Scholar

[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.  Google Scholar

[12]

A. H. KaraF. M. MahomedI. 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.  Google Scholar

[13]

R. E. Lucas Jr., Inflation and welfare, Monetary Theory as a Basis for Monetary Policy, Palgrave Macmillan UK, (2001), 96–142. Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[20]

R. NazF. 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

[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.   Google Scholar

[22]

R. NazF. 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

[23]

R. NazA. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[27]

R. NazF. 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.  Google Scholar

[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.  Google Scholar

[29] L. S. Pontryagin, Mathematical Theory of Optimal Processes, CRC Press, 1987.   Google Scholar
[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.  Google Scholar

[31]

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

[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.  Google Scholar

Figure 1.  Evolution over time of $ c(t), \;u(t), \; k(t) $ and $ h(t) $
Figure 2.  Effect of change of $ \beta $ on evolution over time of $ c(t), \;u(t), \; k(t) $ and $ h(t) $
Table 1.  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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