Modeling economic growth with spatial migration: A stability analysis of the long-run equilibrium based on semigroup theory

We develop a new model of economic growth that fully accounts for the spatial dependence of the flows of workers and capital on salaries and returns. Considering a rather general setting in which we do not require the knowledge of the exact expressions of the production function and the population growth rate, we allow for a strictly positive steady state in which labor, capital, wages, and returns on capital are constant in space. Then, we establish the stability of such an equilibrium using the theory of abstract non-linear parabolic problems and analyzing the spectral properties of the derivative of the operator that describes the coupled dynamics of labor and capital. We present numerical simulations which agree with our theoretical investigation and show that the proposed model allows us to detect interesting transitional dynamics that the standard Solow model does not capture. In particular, we see how the migration of labor can slow down the process of wage convergence at some spatial locations and how the flow of workers, interacting with population dynamics, can reduce economic growth in countries where both developed and underdeveloped regions are present.