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

Rehana Naz

doi:10.3934/dcdss.2020170

Article Contents

2020, Volume 13, Issue 10: 2813-2828. Doi: 10.3934/dcdss.2020170

This issue Previous Article Lie group classification a generalized coupled (2+1)-dimensional hyperbolic system Next Article First integrals of Hamiltonian systems: The inverse problem

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

Rehana Naz^,

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

^* Corresponding author: Rehana Naz
^* Corresponding author: Rehana Naz

Received: January 2019

Revised: May 2019

Early access: December 2019

Published: July 2020

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Abstract

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.

Keywords:

Partial Hamiltonian approach,
unskilled labor.

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

Citation:

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Figure 1. Evolution over time of $ c(t), \;u(t), \; k(t) $ and $ h(t) $

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

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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 $

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