How Does Energy Resource Depletion Affect Prosperity? Mathematics of a Minimum Energy Return on Investment (EROI)

It has been proposed that energy resource depletion and declining energy return on investment (EROI) can disrupt modern, prosperous lifestyles. This is because such lifestyles are dependent on abundant, low-cost energy supplies, to date supplied by fossil energy. We illustrate a mathematical structure by which to analyze the impacts of energy depletion as it affects all sectors of the economy. This framework is based on the reduced availability of discretionary outputs as inter-industry operations become less efficient. We illustrate this mathematical framework, and explore a simple template economy with four sectors. The inputs for each sector are defined at an order-of-magnitude level using data for the US, and the matrix is modified to explore the impacts of resource depletion and uncertainty. We show that the “net energy cliff” concept used in prior studies emerges from the structure of this template economy and appears at similar levels of energy productivity hypothesized in prior work. At levels of net energy return ≤\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\le$$\end{document} 5 J/J, the fraction of productive outputs free to use in discretionary purposes declines rapidly, resulting in the emergence of an effective “minimum EROI” below which prosperity is burdened by excessive direct and indirect requirements of the energy sector. We explore how uncertainty in the matrix specification impacts the level at which the minimum EROI becomes binding. We also show how changes in other sectors (e.g., efficiency of materials production) can affect the rate at which energy depletion affects prosperity.