Mathematical Modeling of Physical Capital Diffusion Using a Spatial Solow Model: Application to Smuggling in Venezuela

This study proposes a mathematical modeling approach for the physical capital diffusion through the borders of countries. Physical capital is considered a crucial variable for the economic growth of a nation. Here, we use an extension of the economic Solow model to describe how smuggling affects the economic growth of countries. In particular, we focus on the situation in Venezuela from 2012 to 2015. In this regard, we rely on a nonconcave production function instead of the classical Cobb–Douglas production function. Moreover, we investigate the effect of different physical capital fluxes on economic growth. The physical capital diffusion through the borders of a country is modeled based on a parabolic partial differential equation describing the dynamics of physical capital and boundary conditions of the Neumann type. Smuggling is present at numerous borders between countries and may include fuel, machinery, and food. This smuggling through the borders places challenges on a particular country’s economy. The smuggling problem usually is related to a non-official exchange rate different from the official rate or subsidies. We study the effect of smuggling on the physical capital of a country using an extended Solow model. Numerical simulations are obtained using an explicit finite difference scheme describing how the physical capital diffusion through the border of a country affects its economic growth. The study of physical capital is a paramount aspect of the economic growth of several countries. The results show that when boundary conditions of Neumann type are different from zero, the dynamics of the physical capital differ from the classical economic behavior observed in the classical spatial Solow model without physical capital flux through the borders of countries. In particular, the numerical results show that the physical capital of a country decreases faster as the flux increases on the boundaries. Thus, we can conclude that avoiding smuggling through the frontiers is a crucial factor affecting economic growth.