A Dynamic Leontief Model with Non-renewable Resources

In this paper we study a generalization of the dynamic Leontief input-output model. We extend the standard dynamic Leontief model with the balance equation of non-renewable resources. Obviously, the non-renewable stocks will decrease, exploiting primary resources. In this study we examine the controllability of this extended model by taking the consumption as the control parameter. Assuming balanced growth for both consumption and production, we investigate how long these scarce resources will cover the input needs of production and how the lifetime of the system depends on the balanced growth rate and on the consumption. In doing so, we apply classic results from control theory and on eigenvalue problems in linear algebra.