Method of Determining Trajectories in a Neighbourhood of Long-Run Equilibrium in Neoclassical Models of Exogenous Economic Growth

Abstract

We consider economic growth models in the form of dynamical systems. We show a method of determining trajectories in a neighbourhood of a long-run equilibrium in some neoclassical models of exogenous economic growth. This method is applied primarily to these models which in general have no analytical solution. We propose the general method of finding solutions of arbitrarily dimensional dynamical system in the form of power series. We expand the state function in Taylor's series in the neighbourhood of the initial state. The coefficients of expansion represent the parameters of the variation of the state of the system and are calculated algebraically in Mathematica. We present the method of finding solutions for the Solow-Swan model and the Mankiw-Romer-Weil model. We use also the Padé aproximant method to obtain a better convergence of the power series. This method allows to obtain a solution in the form of a series for trajectories in a neighbourhood of a long-run equilibrium in two models of exogenous economic growth. We show that obtained solutions are a good approximation of time paths, along which the long-run equilibrium is reached. We show a possibility of estimation of model parameters for which solutions in the form of series are known.

The Solow Model of Economic Growth

“…) or Cattani (2006). Applications of Gaussian hypergeometric functions in the modelling of economic growth processes are discussed in the studies by Ruiz-Tamarit (2004, 2008) or Zawadzki (2015) for the Uzawa-Lucas model, by Guerrini (2006) for the Solow model and by Krawiec and Szydłowski (2002) for the Mankiw-Romer-Weil model.…”