Closed-form solutions of Lucas–Uzawa model with externalities via partial Hamiltonian approach

In this paper, we establish multiple closed-form solutions for all the variables in the Lucas–Uzawa model with externalities for the case with no parameter restrictions as well as for cases with specific parameter restrictions. These multiple solutions are derived with the help of the results derived in Naz et al. (Commun Nonlinear Sci Numer Simul 30(1):299–306, 2016) and Naz and Chaudhry (Math Modell Anal 22(4):464–483, 2017). This multiplicity of solutions is new to the economic growth literature on Lucas–Uzawa model with externalities. After finding solutions for the Lucas–Uzawa model with externalities, we use these solutions to derive the growth rates of all the variables in the system which enables us to fully describe the dynamics of the model. The multiple solutions can potentially explain why some countries economically overtake other countries even though they start from the same initial conditions. We have provided results of the numerical simulations procedure for \(\sigma =\beta \).