Uniqueness and Multiple Trajectories for the Case of Lucas Model

The main aim of this paper is to prove that the model introduced by Lucas and further analyzed by Caballe and Santos, Mulligand and Sala-I-Martin, Benhabib and Perli and finally by Boucekkine and Ruiz-Tamarit, has two interesting properties. If the externality parameter in the production of human capital is greater than the elasticity of output with respect to physical capital, then the system is characterized by multiple transitional paths, indexed by the starting value of the fraction of labor allocated to the production of physical capital, leading to different steady-states equilibrium. Alternatively, if the externality parameter in the production of human capital is smaller than the elasticity of output with respect to physical capital, then the system is characterized by an unique transitional path, convergent to the unique steady-state equilibrium. For the special case where the inverse of the elasticity of intertemporal substitution equals the elasticity of output with respect to physical capital, we obtain closed-form solutions and thus, our approach contains as particular cases some other results, as those obtained by Boucekkine and Ruiz-Tamarit. More than this, differently to Boucekkine and Ruiz-Tamarit we obtain closed-form solutions for all variables of the model and thus we are able to verify the two transitional conditions.