Sustainability of an economy with an exhaustible resource: A viable control approach

We examine the conditions for the sustainability of a production–consumption system based on the use of an exhaustible natural resource. Instead of studying the environmental and economic interactions in terms of optimal control, we focus on the viability of the system, defined by a set of constraints combining guaranteed consumption and a stock of resources to be preserved at all times, which refers to a Rawlsian intergenerational equity perspective. Using the mathematical concept of viability kernel, which makes it possible to deal with the consistency between constraints and controlled dynamics, we exhibit the sustainable technological configurations and, whenever possible, the policy options and environmental-economic states required to obtain a perennial system. We point out the flexibility of the sustainable “extraction–consumption” choices and we show how they are neither reduced to constant consumption paths nor to Hartwick’s rule. Numerical simulations illustrate the general results.