Pollution externalities point to fractal geometry in economics

There has been considerable debate over whether the steady states of economic systems produce fractal geometry. Traditional macroeconomic models have the potential to allow for random dynamics that support fractal sets, and a growing number of studies have set out to characterize these dynamics. By examining the effects of pollution on economics, new work has revealed a potential path forward to understanding fractal geometry within economics.

A team of researchers published new work in Chaos in which they have adapted a standard economic model, the Lucas-Uzawa model, to account for negative pollution externalities. Doing this allowed them to use logarithmic scales to linearize the optimal dynamics and create a fractal typically found in biological or physical structures called the Barnsley fern. The Barnsley fern represents economic steady states within discrete-time dynamical systems. Such fractal geometry is a first within economics.

The team’s most recent paper expands on more than 20 years of work on iterated function systems with probability (IFSP), a linear discrete-time form of dynamical system that attempts to describe optimal dynamics and can be used to approximate fractals.

The group then modified a standard Lucas-Uzawa economic growth model by providing negative coefficients to account for pollution externalities, which yielded negative exponents for production functions within the economy. When log-linearized and using physical and human capital as state variables, the model’s IFSP produced a Barnsley fern fractal, a self-similar structure resembling the structure of a fern in nature.

The authors hope that their findings stoke further interest in exploring fractal geometries in economics and foresee potentially broad-ranging applications for their approach in fields such as epidemiology and wealth distribution.

Source: “Fractal attractors in economic growth models with random pollution externalities,” by Davide La Torre, Simone Marsiglio, and Fabio Privileggi, Chaos (2018). The article can be accessed at https://doi.org/10.1063/1.5023782 .