A spatial version of the AK model of economic growth

Abstract

In this work, we propose a unidimensional spatial generalization of the AK model of economic growth, which is mathematically described by a parabolic linear partial differential equation for the per capita capital of the economy, with corresponding initial and boundary conditions. We obtain Fourier series solutions for the model, considering homogeneous Dirichlet, homogeneous Neumann, and homogenous mixed boundary conditions, and present numerical examples of the model. We show that the model with homogeneous Neumann boundary conditions is a natural spatial generalization for the non-spatial AK model. Besides, we find minimum critical values for the saving rate of the economy that guarantee persistent growth of the per capita capital in the long run, with homogeneous Neumann conditions presenting the lowest value, regardless of the geographical size of the economy, followed by mixed and homogeneous Dirichlet-type conditions, with the minimum value inversely depending on the geographic size of the economy in these last two cases. Finally, the spatial AK model proposed here is an interesting example of application of partial differential equation in the field of Economics.